TEXT-BOOK 


OF 


SYSTEMATIC    MINERALOGY 


HILARY    BAUERMAN,    F.G.S. 


ASSOCIATE   OF    THE    KOVAI.  SCHOOL   Of    WINES 


D.     APPLETON     AND     CO. 

NEW    YORK 

1881 


Sfactf 
Annex 


SOo 
PREFACE. 


IN  preparing  the  present  volume,  two  main  objects  have 
been  kept  in  view  :  first,  that  it  should  form  a  useful  guide 
to  students  desirous  of  acquiring  a  general  knowledge  of  the 
subject ;  and  secondly,  that  it  should  serve  as  an  elementary 
introduction  to  larger  text-books,  such  as  those  of  Dana, 
Miller,  Descloizeaux,and  Schrauf,an  acquaintance  with  which 
is  essential  to  those  who  wish  to  familiarise  themselves  with 
the  higher  branches  of  the  subject.  For  this  purpose,  the 
treatment  adopted  has  been  as  general  as  possible,  the 
descriptions  of  the  crystalline  forms  dealing  only  with  their 
symmetry  and  general  geometrical  properties,  without  enter- 
ing into  the  question  of  the  practical  calculation  and  deter- 
mination of  individual  examples,  which  would  have  increased 
its  bulk  beyond  admissible  proportions.  In  this  part  of  the 
text,  the  methods  followed  have  been  mainly  those  of 
Groth's  admirable  treatise  on  '  Physical  Crystallography,' 
except  that  the  plan  there  adopted  of  considering  the  phy- 
sical structure  of  crystals  before  their  geometrical  properties 
has  been  abandoned  in  favour  of  the  less  logical,  though 
more  familiar,  one  of  giving  precedence  to  the  latter.  The 
optical  properties  of  crystals  have  been  considered  at  some- 
what greater  length  than  is  usual  in  rudimentary  books,  on 
account  of  the  great  and  increasing  use  made  of  this  branch 
of  investigation. 


vi  Preface. 

Upon  similar  utilitarian  considerations  a  mixed  system 
of  notation  has  been  adopted  in  the  crystallographic  part, 
the  forms  being  designated  in  the  text  by  their  symbols 
according  to  Naumann,  while  the  notation  of  their  faces  is 
by  indices  on  Miller's  system.  As  a  matter  of  personal 
preference,  the  latter  system  would  have  been  adopted 
exclusively ;  but,  having  regard  to  the  fact  that  the  former 
is  used  much  more  extensively  than  any  other  system,  both 
in  text-books  and  in  original  memoirs,  familiarity  with  its 
use  is  very  desirable  to  students. 

In  the  hexagonal  system,  the  Bravais-Miller  notation  by 
indices  on  four  axes  has  been  adopted,  as  showing  most 
clearly  the  relation  between  it  and  the  tetragonal  system. 

In  the  chemical  portion  of  the  volume  the  classification 
followed  is  that  of  the  second  edition  of  Rammelsberg's 
'Handbuch  der  Mineral-Chemie,'  as  being  the  standard 
modern  authority  upon  the  chemistry  of  minerals. 

The  systematic  part  having  been  extended  somewhat 
more  than  was  originally  intended,  it  has  been  found  impos- 
sible to  include  physiography,  or  general  descriptive  minera- 
logy, in  the  same  volume,  without  deviating  too  widely  from 
the  plan  of  the  series.  This  will  therefore  be  issued  as  a 
companion  volume. 

In  the  preparation  of  the  work,  valuable  assistance  and 
advice  has  been  received  from  many  friends.  In  gratefully 
acknowledging  these  services,  the  writer  has  to  mention 
particularly  those  rendered  by  the  editor  of  the  series,  Mr. 
Merrifield,  who  has  made  several  important  additions  to  the 
text,  Mr.  R.  T.  Glazebrook,  of  Trinity  College,  Cambridge, 
and  Mr.  F.  W.  Rudler,  who  has  passed  the  later  sheets 
through  the  press  during  the  writer's  absence  abroad. 

LONDON  :  January  10,  1881. 


CONTENTS. 


I.  PRELIMINARY i 

II.  GENERAL  PRINCIPLES  OF  FORM 6 

III.  CUBIC  SYSTEM 37 

IV.  HEXAGONAL  SYSTEM    .        .        .       .        .        .    .  73 

V.  TETRAGONAL  SYSTEM        .        .        .        .  .112 

VI.  RHOMBIC  SYSTEM 128 

VII.  OBLIQUE  SYSTEM      . 145 

VIII.  TRICLINIC  SYSTEM 156 

IX.  COMPOUND  OR  MULTIPLE  CRYSTALS        .        .        .163 

X.  MEASUREMENT  AND  REPRESENTATION  OF  CRYSTALS  190 

XI.  PHYSICAL  PROPERTIES  OF  MINERALS  :— CLEAVAGE, 

HARDNESS,  SPECIFIC  GRAVITY     ....  203 

XII.  OPTICAL  PROPERTIES  OF  MINERALS     ....  218 

XIII.  OPTICAL  PROPERTIES  OF  MINERALS  (conlinited)         ,  280 

XIV.  THERMAL  AND  ELECTRICAL  PROPERTIES  OF  MINERALS  291 
XV.  CHEMICAL  PROPERTIES  OF  MINERALS  .        .        .    .  298 

XVI.  RELATION  OF  FORM  TO  CHEMICAL  COMPOSITION     .  333 

XVII.  ASSOCIATION  AND  DISTRIBUTION  OF  MINERALS        .  345 

INDEX.        ...  363 

5002143 


SYSTEMATIC    MINERALOGY. 


CHAPTER    I. 

PRELIMINARY. 

MINERALOGY  is  the  science  that  treats  of  the  substances 
known  as  Minerals — that  is,  the  constituents  of  the  earth 
considered  as  they  actually  occur  in  nature. 

The  constitution  of  the  solid  earth,  excluding  all  con- 
sideration of  its  inhabitants — that  is,  of  the  animals  and 
plants  living  in  the  atmosphere — may  be  regarded  in  many 
different  ways, 

In  the  most  general  view  the  earth  is  a  spheroid, 
about  five  times  as  heavy  as  an  equal  volume  of  water. 
Geology,  with  somewhat  more  detail,  considers  the  acces- 
sible portion  a  shell  of  some  ten  to  twelve  miles  thick,  as 
made  up  of  about  the  same  number  of  different  kinds  of 
rock,  the  inaccessible  interior  portion  being  probably  not 
very  dissimilar  in  composition;  while  Chemistry  is  con- 
cerned mainly  with  the  ultimate  elementary  constituents  of 
the  mass,  and  the  properties  of  these  elements  as  derived 
from  the  study  of  their  combinations  artificially  formed. 
The  position  of  Mineralogy  is  intermediate  between  those  of 
geology  and  chemistry.  With  the  former  it  considers  the 
structure  of  the  solid  mass  of  the  earth,  but  in  greater  detail, 
resolving  the  rock  masses  into  a  larger  number  of  more 

15 

lA 


2  Systematic  Mineralogy.  [CHAP.  I. 

exactly  defined  constituents  or  minerals ;  while,  with  the 
latter,  it  considers  the  elementary  constitution  of  such 
substances,  restricting  its  study,  however,  to  such  chemical 
compounds  as  are  actually  found  in  nature.  This  indi- 
vidual, natural  existence  is  essential  to  the  idea  of  a 
mineral,  as  distinguishing  it  from  a  chemical  salt  or  other 
artificial  preparation.  Whether  the  latter  is  or  is  not 
represented  in  nature  can  only  be  determined  by  experi- 
ence, but,  speaking  generally,  it  may  be  said  that  only  the 
most  stable  and  least  soluble  compounds,  or  precisely  those 
that  are  most  difficultly  obtainable  in  the  laboratory,  are 
represented  in  nature,  and  therefore  the  chemistry  of 
minerals,  though  essentially  "fragmentary,  is  of  no  small 
importance  in  the  general  body  of  chemical  knowledge. 

The  qualities  essential  to  the  distinction  of  minerals 
among  themselves  are  of  three  kinds — namely,  form,  struc- 
ture, and  chemical  composition — all  of  which  must  be  inves- 
tigated and  determined  before  the  specific  independence  of 
any  mineral  can  be  regarded  as  properly  established.  The 
first  of  these  considers  the  external  form  of  the  substance 
by  methods  which  are  essentially  those  of  descriptive  solid 
geometry,  qualified  by  certain  special  principles — those  of 
symmetry  and  numerical  rationality- — generalised  from  the 
whole  body  of  such  observations.  This  part  of  the  subject 
is  known  as  Geometrical  or  Morphological  Crystallography. 
The  second  quality,  that  of  structure,  considers  the  sub- 
stance as  made  up  of  similar  material  molecules,  whose 
arrangement  is  indicated  by  the  physical  properties,  such  as 
density,  cohesion,  colour,  &c.  ;  or  generally,  by  their  elastic 
resistance  to  forces  tending  to  disturb  their  molecular  equi' 
librium,  which  may  or  may  not  vary  in  different  directions. 
These  investigations  are  essentially  part  of  the  work  of  ex- 
perimental physicists,  but  their  results,  when  combined  with 
those  of  the  geometrical  crystallographer,  collectively  form 
the  branch  of  Physical  Crystallography.  Lastly,  the  inves- 
tigation of  the  third  and  most  important  character,  that  of 


CHAP.  I.]  Definition  of  Species.  3 

elementary  composition,  is  the  work  of  Mineral  Chemistry  ; 
and  the  combination  of  this  with  the  knowledge  derived  from 
the  study  of  form  gives  rise  to  the  important  principle  of 
Isomorphism,  or  the  relation  of  form  to  constitution,  upon 
which  the  most  natural  and  satisfactory  systems  of  classifi- 
cation are  founded. 

In  addition  to  these  three  principal  heads,  information 
upon  subsidiary  matters  is  requisite  for  the  attainment 
of  a  complete  knowledge  of  any  mineral.  These  have  re- 
ference to  its  natural  habitat,  such  as  association  with  other 
minerals,  geographical  and  geological  distribution,  and  evi- 
dences of  possible  changes  from  the  condition  of  original 
formation.  This  last  point  especially  has  an  important 
bearing  upon  the  speculative  matter  of  the  origin  and  mode 
of  formation  of  minerals,  which  is,  or  should  be,  the  province 
of  geology  proper,  as  the  basis  of  any  reasonable  speculation 
upon  the  structure  of  the  earth  in  its  largest  sense.  This 
branch  of  the  subject  is,  however,  generally  spoken  of  as 
Chemical  Geology. 

The  whole  body  of  knowledge  comprised  under  these 
several  heads,  when  classified  in  an  orderly  manner,  forms 
Descriptive  Mineralogy,  or  Physiography. 

The  number  of  elements  known  to  the  chemist  is  now 
between  sixty  and  seventy,  all  of  which  are  concerned  in 
the  production  of  minerals,  though  in  widely  different  pro- 
portions. Some  eight  or  ten  are  found  in  the  free  or  un- 
combined  state,  and  with  six  or  seven  hundred  combinations 
of  two  or  more,  make  up  the  roll  of  natural  minerals  or 
mineral  species. 

The  term  species  is  applied  to  any  substance  whose 
form,  structure,  and  composition  are  definite,  constant,  and 
peculiar  to  itself,  and  therefore  serve  to  distinguish  it  from 
all  other  species.  By  constancy  of  form  and  composition 
in  the  above  definition  is  not  meant  that  there  must  be 
absolute  identity  in  these  particulars  between  different  ex- 
amples of  the  same  substance,  but  that  the  variations  in 
B  2 


4  Systematic  Mineralogy.  [CHAP.  L 

either  shall  be  subject  to  known  laws.     Thus,  carbonate  of 
calcium  in   the  species  Calcite  appears  in  several  hundreds 
of  different  forms,  all   of  which  may,  by  the  application 
of  crystallographic  laws,  be  shown  to  be  derivatives  of  a 
single  geometrical  form — the  rhombohedron— and  again  its 
composition  may  differ  sensibly  in  different  specimens,  but 
these  differences  are  explainable  by  the  law  of  isomorphism, 
which  shows  that  one  dyad  metal  may  be  substituted  for 
another  without  altering  the  general  molecular  constitution. 
The  exact  limits  to  be  given  to  species  is  to  a  great 
extent  matter  of  opinion.     Upon  purely  chemical  grounds, 
all  substances  combining  the  same  type  of  molecular  consti- 
tution with  analogous  forms  may  be  considered  as  varieties 
of  a  single  species,  without  reference  to  the  nature  of  the 
elements  composing  them ;  but  this  is  too  wide  a  definition 
to  be  of  much  practical  use  to   the   mineralogist.      It  is 
therefore  customary  with  such  a  class  of  similarly  constituted 
substances  to  classify  them   according  to  their  contained 
metals,  giving  a  different  name  to  each,  and  to  call  the 
whole  group  by  the  name  of  the  most  prominent  species. 
On  the  other  hand,  a  variety  of  any  substance  marked  by  some 
constant  peculiarity,  whether  of  form,  colour,  or  other  ap- 
parent property,  may  often  receive  a  particular  name  with  ad- 
vantage, even  when  the  distinguishing  difference  is  too  slight 
to  allow  of  the  separation  on  the  grounds  of  systematic  form 
or  composition.     The  ultimate  guide  is  in  all  cases  the  con- 
venience of  the  observer ;  and  if  the  practice  of  giving  names 
without  a  previous  complete  determination  of  the  compo- 
sition and  physical  properties  be  avoided,   it  is  generally 
better  to  form  new  specific  names  rather  than  unduly  widen 
the  boundaries  of  older  ones. 

The  proportion  in  which  different  minerals  enter  into 
the  composition  of  the  crust  of  the  earth  varies  very  con- 
siderably, as  does  also  the  size  of  the  individual  masses 
of  any  one.  Thus  we  may  find  the  same  substance  in 
particles  of  microscopic  minuteness  in  some  places,  and  in 


CHAP.  I.]  Rocks  and  Minerals.  5 

others  in  masses  measurable  by  cubic  feet  or  yards,  and 
weighing  up  to  hundreds  of  tons,  or  even  forming  mountain 
masses,  without  sensible  admixture  of  other  substances. 
Experience,  however,  shows  that  the  characteristic  proper- 
ties of  any  mineral,  and  especially  that  of  form,  are  best 
developed  with  individuals  of  a  moderate  size,  as  when  very 
minute  they  become  invisible  and  incapable  of  exact  mea- 
surement, and  when  very  large  the  characteristic  form  is  not, 
as  a  rule,  apparent.  Such  undefined  mineral  masses  ar"e 
generally  spoken  of  as  Rocks.  Quartzite  and  statuary 
marbles,  for  instance,  are  aggregates  of  particles  of  quartz 
and  calcite  into  masses  of  a  slaty  or  granular  texture  in 
which  their  proper  forms  are  entirely  lost  ;  while,  on  the 
other  hand,  the  crystalline  lava  called  basalt  is  made  up  of 
individuals  of  the  species  Labradorite,  Augite,  Olivine,  and 
Magnetite,  all  perfectly  well  defined  in  form  and  physical 
characters,  but  so  minute  as  to  be  indistinguishable  to  the 
unaided  eye,  the  general  effect  being  that  of  a  uniform, 
opaque,  black  substance,  separating  into  masses  whose  shapes 
bear  no  obvious  relation  to  those  of  their  constituent  minerals. 
The  distinction  between  rocks  and  minerals  is,  however,  one 
of  geological  convenience  only,  and  in  mineralogy  is  almost 
without  significance  ;  the  nature  of  a  mineral  mass  being 
defined  according  to  its  constituents,  either  as  of  a  single 
species  or  an  aggregate  of  two  or  more,  without  reference  to 
the  size  or  perfection  of  the  individual  components.  It 
often  happens  that  groups  of  two  or  more  minerals,  dis- 
tinctly separated,  pass  in  the  same  mass  by  inappreciable 
gradations  into  aggregates  in  which  the  individuals  are  indis- 
tinguishable by  ordinary  means,  so  that  it  is  difficult  to  say 
where  either  condition  begins  or  ends.  This  difficulty  is 
further  increased  by  the  use  of  the  microscope,  which  very 
commonly  resolves  substances  apparently  uniform  into  aggre- 
gates of  dissimilar  ones,  and  there  is  no  reason  to  suppose 
that  the  individuality  of  the  constituents  ceases  when  the 
microscope  is  no  longer  able  to  reveal  them.  The  same 


6  Systematic  Mineralogy.  [CHAP.  II. 

class  of  observation,  however,  shows  that  foreign  substances 
are  so  commonly  included  even  in  the  most  perfectly  deve- 
loped mineral  individuals  or  cr)  stals,  that  the  condition  of 
homogeneity  required  by  the  ordinary  definition  of  a 
mineral  as  a  homogeneous  inorganic  substance  is  seldom,  if 
ever,  realised,  and  therefore  this  definition  can  only  be 
accepted  as  an  approximation  requiring  considerable  qualifi- 
cation in  use. 

The  complete  discussion  of  all  the  subjects  indicated  in 
the  preceding  pages,  or  even  of  any  one  of  them,  being 
beyond  the  scope  of  an  elementary  book  of  limited  extent, 
the  space  at  command  will  be  devoted  to  a  sketch  of  the 
principles  upon  which  the  methods  of  determining  form, 
structure,  and  other  elements  of  classification  in  minerals  are 
based,  without  entering  into  the  details  of  the  methods  of 
observing  or  reduction  of  observations,  for  which  matters 
the  student  is  referred  to  the  larger  special  works  as  given 
in  the  list  at  the  end  of  the  volume.  A  physiographic  sketch 
of  the  more  important  species,  classified  according  to  Ber- 
zelius'  and  Rammelsberg's  method,  forms  the  subject  of  a 
companion  volume. 


CHAPTER   II. 

GENERAL    PRINCIPLES    OF    FORM. 

WITH  the  exception  of  water,  mercury,  and  some  hydro- 
carbons which  are  liquids  at  ordinary  temperatures,  minerals 
are  solids,  and  occur  in  masses  which  in  some  cases  are  of 
irregular,  and  in  others  of  regular,  shape.  The  first  of  these 
are  called  amorphous,  and  the  second  crystalline,  substances, 
and  any  individual  mass  of  the  latter  kind  is  a  crystal.  The 
branch  of  mineralogy  that  treats  of  the  study  of  such  form 
is  called  Crystallography. 


CHAP.  II.]  Definition  of  Symmetry. 

The  term  '  crystal,'  derived  from  the  Greek  cpu 
which  was  applied  to  ice  and  transparent  quartz  or  rock- 
crystal,  the  latter  having  been  supposed  to  be  produced 
from  water  by  extreme  cold  in  mountain  regions,  is  ap- 
plied to  natural  and  artificial  substances  which,  in  solidi- 
fying, from  a  state  whether  of  solution-  or  fusion,  assume 
definite  polyhedral  forms,  which  are  constant  for  the  same 
substance.  The  leading  property  of  crystals,  as  distinguished 
from  mere  geometrical  solids,  is  the  invariability  of  the 
angles  between  corresponding  faces  in  different  individuals 
of  the  same  substance.  There  is  usually  a  very  marked 
symmetry  to  be  noticed  in  the  arrangement  of  their  plane 
faces  and  edges,  and  occasionally  of  their  points  also, 
although  this  latter  symmetry  is  not  essential,  crystallo- 
graphic  symmetry  being  one  of  direction  and  not  of  position, 
so  that  two  parallel  planes  or  two  parallel  lines  are  not 
distinguished  from  one  another,  and  on  that  account  the 
invariability  of  the  angles  is  a  paramount  consideration. 
The  character  of  the  symmetry  varies  in  different  groups  of 
crystals,  and  forms  the  basis  of  their  classification  into 
systems. 

If  we  take  any  polyhedron  and  place  it  upon  a  looking- 
glass,  the  object  and  its  image  together  constitute  a  sym- 
metrical figure,  of  which  the  reflecting  surface  is  the  plane 
of  symmetry  ;  and  if  in  any  given  solid  we  can  find  a  plane 
such  that,  if  we  were  to  cut  it  in  half  by  that  plane,  and  to 
place  against  the  section  a  mirror,  the  reflected  image 
would  exactly  reproduce  the  other  half  (identically,  and  not 
reversed,  as  objects  generally  are  by  reflection),  the  solid 
is  said  to  be  symmetrical  about  that  plane.1  This  is  a 

1  This  may  be  shown  with  a  model  of  a  cube,  painted  white,  and  a 
plate  of  red  glass,  held  perpendicularly  to  one  of  its  faces,  when  a  white 
image  of  the  part  of  the  face  in  front  will  be  seen  by  reflected,  and  a 
red  one  of  the  hinder  part  by  transmitted,  light.  If  the  direction  of  the 
plate  be  parallel  to  that  of  a  plane  of  symmetry  the  two  images  will 


Systematic  Mineralogy. 


[CHAP.   II. 


more  particular  supposition  than  need  be  made  in  crystallo- 
graphy, in  which  two  parallel  planes  are  not  distinguished  ; 
nevertheless  it  is  convenient  to  use  it  for  purposes  of  de- 
scription, in  order  to  escape  a  vagueness  which  would  other- 
wise make  the  description  unintelligible.  For  instance, 
although  figs,  i  and  2  have  exactly  the  same  crystallo- 
graphic  symmetry,  it  will  be  more  convenient  to  consider 
and  describe  the  first,  and  to  regard  the  second  as  another 
example  of  the  same  form.  They  are  both  symmetrical,  as 


FIG.  i. 


FIG.  2. 


regards  direction,  to  the  lines  A B,  CD,  but  the  symmetry  of 
position  of  fig.  i  renders  it  a  much  more  definite  thing 
to  talk  about  and  to  apply  linear  measure  to.  Only  it  must 
not  be  forgotten  that  this  symmetry  of  position  is  neither 
essential  nor  inherent,  but  is  merely  adopted  as  an  aid  in 
forming  definite  ideas. 

Symmetry  about  a  line  in  plane  figures  corresponds  to 
symmetry  about  a  plane  in  space.  In  the  latter,  symmetry 
about  lines  or  axes  is  also  observed  in  many  cases,  but  the 
discussion  of  this  may  be  deferred  until-  another  character- 
istic feature  of  crystals,  and  that  the  most  important  one, 
as  forming  the  basis  of  all  exact  crystallography — namely, 
the  principle  of  rationality— has  been  noticed. 

The  term  '  Rationality '  will  be  best  understood  by  using 

apparently  coincide  ;  but  in  any  other  position  they  will  deviate  to  a 
greater  or  less  extent.  The  particular  positions  of  symmetry  for  the 
cube  are  shown  in  figs,  n,  12,  13. 


CHAP.  II.]  Principle  of  Rationality.  9 

plane  space  of  two  dimensions,  instead  of  actual  space  of 
three  dimensions,  for  an  illustration.     Without  considering 
the  exact  meaning  of  axes  of  refer- 
ence, let  it  be  assumed  that  o  A,  o  B, 
fig.   3,  represent  two  such  axes,  to 
which  two  planes  belonging  to  one 
crystal,  represented  by  P  Q,  P'  Q',  are 
referred.     Then  the  principle  of  ra- 
tionality   requires    that    if  PQ"   be 
drawn  through  P  parallel  to  P'  Q',  the 
ratio  of  o  Q  to  o  Q"  shall  always  be  rational,  or,  as  it  may  be 
more  generally  stated, 

OP       O  "P 

-  =  a  simple  rational  fraction. 
OQ    OQ' 

Usually,  the  relation  is  one  of  very  simple  numbers,  such  as 
2  :  T,  i  :  2,  2  :  3,  4  :  5,  &c.,  while  it  can  never  be  that  of 
an  incommensurable  surd,  such  as  \/2  or  \/5,  to  unity. 
This  law  is  an  empirical  one — that  is,  it  expresses  the  results 
of  observations  without  explaining  their  cause — but  there  are 
no  known  exceptions.  Its  geometrical  consequences  are — 

1.  The  exclusion  of  all  but  the  simpler  types  of  sym- 
metry about  an  axis,  namely,  binary,  quaternary,  ternary, 
and  senary. 

2.  The  exclusion,  from  possible  crystalline"  forms,  of  the 
Platonic  or  regular  geometrical  solids  of  higher  order  than 
the  cube  or  octahedron. 

The  regular  dodecahedron  and  icosahedron  involve 
pentagonal  symmetry,  and  they  bring  the  irrational  value \/ 5 
into  the  axial  relations.  There  are  numerous  examples 
among  natural  crystals,  notably  in  iron  pyrites,  of  forms  ap- 
proximating in  shape  to  these — that  is,  they  are  contained  by 
twelve  five-sided  or  twenty  three-sided  faces — but  these  are 
never  regular  pentagons,  or  all  equilateral  triangles.  When 
applied  to  three  dimensions,  as  in  actual  crystals,  the  prin- 
ciple of  rationality  requires  that  if  planes  of  different  kinds 


io  Systematic  Mineralogy.  [CHAP.  II.. 

occur  in  the  same  crystal  they  must  be  so  related  that  their 
intercepts *  upon  the  axes  of  reference  are  in  rational  pro- 
portion to  one  another.  That  is  to  say,  if  one  plane  meets- 
the  three  axes  in  the  points  pqr,  and  the  other  in  the  points 
p  Q  R,  the  relation  expressed  by 

op    o  q    o  r 
OP ' OQ " OR 

will  be  that  of  quantities  having  rational  ratios  to  one  another,, 
and  usually  these  will  be  found  to  be  in  low  numbers. 

Axes  of  symmetry.  When  a  polyhedron  is  turned  about 
a  line  so  selected  that  after  passing  through  an  aliquot  part 
of  a  whole  revolution  its  position  in  space  as  a  whole  has 
not  been  changed,  and  there  is  no  apparent  difference  in 
shape  for  the  two  aspects,  it  is  said  to  be  symmetrical  to  the 
line  or  axis  of  rotation.  The  kind  of  symmetry  is  denoted 
by  the  number  of  times  the  positions  of  symmetry  recur  in 
a  complete  revolution,  or  between  the  starting  of  a  marked 
point  on  the  crystal  and  its  return  to  the  original  position. 
Thus,  in  a  cube,  the  straight  line  joining  the  middle  points  of 
two  opposite  edges  is  an  axis  of  binary  symmetry  ;  for  by 
turning  the  solid  about  such  a  line  through  half  a  revolution, 
it  assumes  a  position  apparently  similar  to  the  original  one, 
and  the  change  can  only  be  perceived  by  observing  the  alte- 
ration in  the  place  of  a  marked  point  or  face  with  reference 
to  some  external  object.  The  cube  has  also  ternary  sym- 
metry about  a  diagonal  line  joining  opposite  points,  and 
quaternary  symmetry  about  a  line  joining  the  centres  of 
opposite  faces,  the  original  position  being  apparently  re- 
stored by  rotation  through  one  third  of  a  revolution  in  the 
former,  and  one  quarter  or  one  right  angle  in  the  latter  case. 
Quaternary  also  includes  the  lower  condition  of  binary 

1  An  intercept  is  the  distance  intercepted  or  cut  off  by  a  plane  upon: 
an  axis  measured  from  the  origin  of  the  latter.  Thus  in  fig.  3,  OQ, 
O  Q',  O  P,  O  P',  are  intercepts  upon  the  axes  o  B,  o  A,  whose  origin  is  at  O, 


CHAP.  II.]  CrystallograpJiic  Systems.  II 

symmetry.  The  only  other  class  of  symmetry  possible  in 
crystals  is  senary  or  hexagonal,  corresponding  to  a  rotation 
of  one-sixth  of  a  revolution,  such  as  that  of  a  regular  hexa- 
gonal prism  about  its  axis ;  this  includes  ternary  symmetry. 
Quinary  symmetry,  such  as  that  of  a  Platonic  or  regular 
icosahedron  about  a  diagonal,  or  of  a  Platonic  dodecahedron 
about  a  line  joining  the  centres  of  opposite  faces,  is  crys- 
tallographically  impossible,  as  it  introduces  irrational  re- 
lations. The  remarks  made  on  page  8  apply  equally  in 
this  case :  the  necessary  symmetry  being  one  of  direction 
only,  the  same  symmetry  exists  about  any  line  parallel  to 
the  axis  as  about  the  axis  itself;  but  for  convenience  of 
description  it  is  best  to  consider  the  cases  in  which  there  is. 
also  symmetry  of  position,  always  bearing  in  mind  that  this 
is  a  mere  matter  of  convenience,  and  not  essential,  or 
affecting  the  classification. 

As  has  already  been  shown  in  the  case  of  the  cube,  a 
crystal  may  be  symmetrical  about  more  than  one  axis.  If 
there  is  binary  or  quaternary  symmetry  about  two  axes  at. 
right  angles  to  one  another  there  is  a  third  axis  of  the  same 
kind  at  right  angles  to  both. 

Crystals  are  classified  into  systems  according  to  the 
number  and  character  of  their  axes  of  symmetry.  Six  of 
such  systems  are  possible,  all  of  which  are  represented  by 
natural  minerals.  They  are  as  follows,  commencing  with 
those  of  lowest  symmetry  : — 

1.  The  triclinic  system.     This  is  without  any  axis   of 
symmetry,  the  faces  of  any  form  being  only  symmetrical  to 
a  central  point. 

2.  The  oblique  system.     This  has  one  axis  of  binary 
symmetry,  and  consequently  one  plane  of  symmetry. 

3.  The  rhombic  system.     This  has  three  axes  of  binary 
symmetry  at  right  angles  to  one  another. 

4.  The  hexagonal  system.     This  is  characterised  by  one 
axis  of  senary  or  hexagonal   symmetry  and  six  of  binary 
symmetry  at  right  angles  to  the  first.     It  includes  the  case 


12  Systematic  Mineralogy.  [CHAP.  II. 

in  which  there  is  an  axis  of  ternary  symmetry,  or  the  rhom- 
bohedral  system. 

5.  The  tetragonal  system.     This  has  one  axis  of  qua- 
ternary symmetry  at  right  angles  to  two  of  binary  symmetry, 
which  are  also  at  right  angles  to  each  other. 

6.  The  cubic  system.     This  is  specially  characterised 
by  three  axes  of  quaternary  symmetry  at  right  angles  to 
•one  another,  besides  which  there  are  four  of  ternary  and 
•six  of  binary  symmetry,  whose  positions  have  been  already 
alluded  to  on  p.  10,  and  will  be  more  specially  noticed  sub- 
.sequently. 

When  a  system  has  more  than  one  kind  of  symmetry,  it 
may  be  distinguished  by  the  number  of  its  axes  of  the 
highest  kind,  or  axes  of  principal  symmetry.  Upon  this  dis- 
tinction is  founded  the  classification  of  the  systems  into  the 
following  three  groups,  which  are  closely  related  to  their 
physical  properties  : — 

1.  Without  a  principal  axis  of  symmetry.     This  includes 
the  triclinic,  oblique,  and  rhombic  systems,  the  first  being 
without  linear  symmetry,  while  in  the  second  and  the  third 
the  symmetry  is  all  of  the  same  kind,  namely,  binary. 

2.  With  one  principal  axis  of  symmetry.     This  includes 
the  hexagonal  and  tetragonal  systems,  the  principal  sym- 
metry of  the  first  being  senary  and   of  the  second   qua- 
ternary. 

3.  With   three  principal    axes    of  symmetry.     This   is 
special  to  the  cubic  system,  the  three  axes  being  those  of 
quaternary  symmetry. 

When  a  crystal  is  contained  by  all  the  planes  or  faces  ' 
required  by  the  complete  symmetry  of  the  system,  each  one 
has  a  counterpart  plane  parallel  to  it,  so  that  their  total 
number  is  always  even  and  not  less  than  six.  These  are 

1  It  is  convenient  to  call  the  natural  surfaces  of  crystals  faces,  and 
those  produced  artificially,  or  required  in  geometrical  construction, 
planes,  e.g.  planes  of  symmetry  and  cleavage  planes. 


CHAP.  II.]  Hemihcdrisin.  13 

said  to  be  holohedral  (full-faced)  forms.  There  are  also 
certain  forms  in  which  only  one  half  or  one  quarter  of  the 
full  number  of  faces  are  present ;  these  are  respectively 
called  hemihedral  (half-faced)  and  tetartohedral  (quarter- 
faced)  forms.  The  selection  of  these  faces  may  in  some 
instances  be  made  in  more  ways  than  one,  subject  to  the 
condition  that  the  relation  of  the  faces  to  the  axes  of  sym- 
metry must  be  the  same  as  in  the  holohedral  form,  or  each 
equivalent  axis  must  cut  an  equal  number  of  faces — namely, 
one  half  or  one  quarter  of  that  which  it  would  do  at  the 
same  inclination  in  the  full-faced  form. 

Hemihedral  forms  are  not  possible  in  the  triclinic 
system,  that  being  without  plane  symmetry  ;  in  the  oblique 
system  there  may  be  one  kind,  but  it  has  not  been  observed 
in  nature  ;  in  the  rhombic  there  may  be  two  kinds,  but  it  is 
doubtful  whether  more  than  one  actually  exists  in  nature. 
In  the  remaining  systems,  hexagonal,  tetragonal,  and  cubic, 
the  forms  are  susceptible  of  hemihedral  development  in 
three  ways,  but  it  is  only  in  the  tetragonal  that  all  three  are 
actually  known  to  exist,  and  of  these  one  has  only  been 
observed  in  artificial  organic  compounds. 

Tetartohedral  crystals  are  possible  in  all  systems  where 
there  are  more  than  two  kinds  of  hemihedrism,  that  is  in 
the  tetragonal,  hexagonal,  and  cubic.  As  they  may  be  con- 
sidered as  resulting  from  the  successive  application  of  two 
kinds  of  hemihedry  to  holohedral  forms,  three  classes  of 
tetartohedra  might  be  possible  in  the  first  two  systems,  were 
it  not  for  the  circumstance  that  in  either  only  two  out  of 
the  three  combinations  give  rise  to  forms  that  satisfy  the 
general  conditions  of  symmetry.  In  the  cubic  system  all 
•  three  kinds  of  hemi-hemihedrism  produce  the  same  class 
of  form  or  there  is  only  one  kind  of  tetartohedron.  This 
has  not  been  found  in  natural  minerals,  but  is  characteristic 
of  a  group  of  metallic  salts,  the  nitrates  of  the  lead-barium 
group.  In  the  tetragonal  and  hexagonal  systems  the  two 
possible  methods  give  rise  to  two  different  kinds  of  forms, 


14  Systematic  Mineralogy.  [CHAP.  II. 

neither  of  which  has  been  observed  either  in  natural  or 
artificial  crystals  in  the  former  system,  but  in  the  latter  both 
kinds  are  known,  one  of  them  being  specially  characteristic 
of  the  commonest  mineral  constituent  of  the  earth's  crust, 
namely,  quartz  or  rock-crystal. 

By  hemihedral  or  tetartoheclral  development  a  crystal 
belonging  to  any  system  loses  a  portion  of  the  symmetry 
which  characterises  it  when  possessed  of  the  full  number  of 
faces.  Thus,  in  the  three  kinds  of  hemihedra  possible  in  the 
systems  with  principal  axes,  one  is  entirely  without  plane 
symmetry,  while  each  of  the  other  two  has  only  one  of  the 
two  kinds  necessary  in  the  full-faced  forms.  It  must  there- 
fore be  remembered  that  in  defining  systems  by  their  charac- 
teristic symmetry,  such  definitions  only  apply  to  the  holo- 
hedral  forms. 

There  is  an  essential  distinction  between  the  geometrical 
and  the  mineralogical  idea  of  hemihedrism  and  tetartohe- 
drism,  which  it  may  be  well  to  notice  as  early  as  possible.  In 
the  former  only  those  forms  are  considered  as  hemihedra  and 
tetartohedra  that  are  geometrically  distinguishable  from  the 
holohedral  forms,  while  in  the  latter  the  presence  of  any 
such  form  in  any  crystal  of  a  substance  is  considered  as  im- 
parting the  same  character  to  all  other  crystals  of  the  same 
substance,  whether  they  be  geometrically  distinguishable 
from  holohedral  forms  or  not. 

The  same  kind  of  symmetry  as  is  displayed  in  crystals 
is  furnished  by  an  orderly  arrangement  of  points  in  space, 
which  has  the  further  analogy  with  crystals  of  suggesting 
rational  ratios.  These  arrangements  of  points  may  be  ex- 
tremely simple  ;  indeed  all  the  crystallographic  systems  may 
be  represented  by  the  points  of  intersection  of  three  sets  of 
parallel  and  equidistant  planes,  as  in  fig.  4,  where  we  may 
call  a\a±  =  a,  a^a*  =  b,  and  a^  =  c,  and  the  angles  a-aaj)^  =  o, 
a}a4l>4  =  /3,  and  a{a4a5  =  y.  The  whole  series  of  points  may 
be  regarded  as  a  sort  of  net  in  space,  of  which  the  strings 
represent  the  lines  of  intersections  of  these  planes,  and  the 


•CHAP.  II.] 


Reticular  Point  Systems. 


3 


--!-"•«• 

;->«*    ° 


knots  or  nodes  the  points  of  intersection.  Then,  if  the 
equal  distances  at  which  one  set  of  planes  intersects  the 
other  two  be  taken  as  a,  b,  and  c  respectively,  and  the 
angles  between  the  lines  as  a,  /3,  FIG.  4. 

and  y,  these  quantities  will  be 
the  characteristics  of  the  system, 
and  symmetrical  relations  of 
equality  between  those  charac- 
teristics will  determine  a  crystal- 
lographic  system.  The  planes 
of  the  system,  corresponding  to 
faces  of  the  crystal,  will  be  planes 

drawn  through  any  three  points  of  the  system.  The  limita- 
tion to  rational  ratios  is  at  once  suggested  by  the  necessity  of 
taking  a  whole  number  of  intervals  between  any  two  points,  in 
order  to  satisfy  the  definition  of  a  plane  of  the  system  being 
•one  passing  through  three  points  of  the  system.  As  to  the 
particular  modes  of  symmetry,  if  a  /3  y  are  all  different, 
there  is  no  symmetry.  This  corresponds  to  the  anorthic  or 
triclinic  system.  If /Bandy  are  right  angles,  a  is  perpen- 
dicular to  the  plane  of  be,  and  is  an  axis  of  binary  sym- 
metry. We  have  then  the  oblique  system.  If  a  /3  y  are  all 
right  angles,  we  have  three  axes  of  binary  symmetry,  cor- 
responding to  the  rhombic  system.  If,  in  addition,  b  =  c, 
•a  is  an  axis  of  quaternary  symmetry,  and  we  obtain  the 
tetragonal  system  ;  while  if  a  =  b  =  c  we  have  the  cubic 
system.  If,  on  the  other  hand,  «=/3  =  y,  without  being 
right  angles,  and  also  a  =  b  =  c,  then  there  is  an  axis  of 
ternary  symmetry  equally  inclined  to  a,  b,  and  c.  This 
represents  the  rhombohedral  system  of  ternary  symmetry 
which  has  been  mentioned  as  included  in  the  hexagonal 
•system. 

It  is  to  be  remarked  that  while  these  relations  between 
a  b  c,  a  /3  y,  yield  the  crystallographic  systems  in  the  easiest 
way,  they  are  sufficient,  but  not  necessary.  For  instance,  if 
a  j3  y  are  all  right  angles,  it  is  not  necessary  that  a,  b,  and  c 


1 6  Systematic  Mineralogy,  [CHAP.  II, 

should  be  all  equal  in  order  to  give  a  cubic  system  :  it  is 
sufficient  that  they  should  be  commensurable.  For,  let 
a  —  i,  &=•  2,  ^=3,  then  a  cube  whose  side  is  6  will  include 
all,  and  will  thus  yield  cubic  symmetry.  There  may  also  be 
particular  relations  between  a  b  c,  o  /3  y,  of  a  less  simple 
character,  which  will  afford  a  similar  increase  of  symmetry. 
This  will  be  at  once  seen  by  joining  three  points  of  a 
cubical  system,  taken  arbitrarily,  to  a  fourth.  If  we  com- 
plete the  parallelepiped,  and  take  its  planes  as  the  basis  of 
a  new  system  of  points,  the  old  axes  will  no  longer  be 
edges  of  the  fundamental  parallelepiped,  but  they  will  not 
thereby  cease  to  be  axes  of  symmetry,  for  it  will  always  be 
possible  to  find  planes  of  the  new  system  (that  is,  each 
passing  through  three  points  of  it)  which  shall  be  at  right 
angles  to  them.  As  a  particular  example,  if  a-=b  =  c,  and 
a  =  /3  — y,  we  have  in  general  the  rhombohedral  system; 
but  if  a  takes  the  particular  value,  70°  31' 44",  so  that  the 
dihedral  angles  are  of  60  degrees,  this  introduces  further 
symmetry,  and  throws  us  back  on  the  cubic  system. 

The  conditions  on  which  an  arbitrary  parallelepipedal 
system  should  have  a  given  symmetry  are  not  fully  known, 
and  must  at  any  rate  be  extremely  complicated.  In  place 
of  attempting  any  such  reductions,  it  will  be  preferable  to 
work  from  known  axes  of  symmetry  in  each  system. 

Notation  of  a  plane  face.  In  all  the  systems,  the  position 
of  a  face  of  a  crystal  is  represented  by  the  points  at  which  it 
cuts  the  three  selected  axes.  Since,  moreover,  the  ratios 
of  the  intercepts  must  always  be  commensurable,  this  is 
secured  by  taking  the  intercepts  as  integral  multiples  (or  sub- 
multiples)  of  definite  lengths  measured  along  each  axis,  and 
called  the  parameters  or  units.  These  lengths  are  not  arbi- 
trary, but  depend  upon  the  nature  of  the  substance  forming 
the  crystal.  If  we  consider  this  as  composed  of  a  series  of 
molecules  in  parallelepipedal  order,  it  secures  that  the  faces 
shall  be  planes  of  the  system. 

Notation  of  faces  of  crystals.     There  are  four  principal 


CHAP.  II.] 


Weiss' s  Notation. 


methods  of  indicating  the  faces  of  crystals  by  their  axial  rela- 
tions in  use — those  of  Weiss,  Miller,  Naumann,  and  Le'vy  ; 
and  as  all  are  of  equal  authority,  it  will  be  necessary  for  the 
student  to  become  acquainted  with  the  principles  of  each. 
The  last  of  these  is  a  modification  of  the  oldest  system,  that 
of  the  Abbe'  Haiiy,  which  was  in  the  earlier  years  of  this 
century  in  common  use  by  mineralogists  in  most  European 
countries ;  but  during  the  last  quarter  of  a  century  it  has 
in  many  cases  been  superseded  by  one  or  other  of  the 
remaining  systems,  none  of  which  have,  however,  up  to  the 
present  time  become  sufficiently  popular  to  be  universally 
accepted.  In  the  first  two  methods  a  face  is  indicated  by  a 
symbol  compounded  of  three  signs  derived  from  the  ratios 
of  the  axial  intercepts  to  the  unit  parameters — that  is,  one 
for  each  axis — which  are  written  in  an  invariable  order ;  while 
in  the  other  two,  arbitrary  signs,  which  vary  with  the  systems, 
are  used  in  addition. 

Weiss's  notation.  In  this,  the  least  conventional  of  all 
the  methods,  a  unit  face  A  B  c  is  indicated  by  the  length  of  its 
intercepts  upon  the  axes  of  reference  in  the  order  shown  in 
fig.  5,  as  a  :  b  :  c,  where  o  A  =  a, 
o  B  =  £,  o  c  =  c.  This  supposes 
the  parameters  to  be  dissimilar  for 
all  three  axes,  as  in  the  rhombic, 
oblique,  and  triclinic  systems ; 
but  when  a  =  b,  as  in  the  hexa- 
gonal and  tetragonal  systems, 
the  symbol  used  is  a  :  a  :  c,  and 
in  the  cubic  system,  where  all 
three  parameters  are  alike,  it  is 
a  :  a  :  a.  For  any  face  having  different  intercepts,  such 
as  A  K"  c',  the  longer  ones  are  expressed  as  multiples  of 
the  shortest  considered  as  unity ;  or  for  the  particular 
case  where  the  intercept  OB"  is  i^  times  OB,  and  o  c' 
3  times  oc,  a  :  f  b  :  3  c,  all  the  signs  being  positive  ;  while 
for  a  parallel  face  placed  behind,  to  the  left,  and  below 

c 


FIG.  5. 


1 8  Systematic  Mineralogy  [CHAP.  n. 

the  point  o,  the  same  symbol  is  used,  hut  with  negative 
signs  or  accented  letters  thus,  —  a  :  —  -i] b  :  —  3  c,  or 
a'  :  f  b'  :  $<:',  the  co-efficients  being  any  rational  whole 
number  or  fraction,  including  o  and  oo.  For  a  face  such 
as  A  F  B,  where  o  F  =  ^  o  c,  the  symbol  is  a  :  b  :  ^  c.  When 
one  of  the  intercepts  is  immeasurably  long,  its  co-efficient 
becomes  oo,  as  in  the  face  A  B"  D  E,  which  is  parallel  to  the 
vertical  axis  c,  the  symbol  is  a  :  %b  :  oo  c. 

The  general  expression  for  any  face  meeting  the  three 
axes  at  dissimilar  distances  is  a  :  nb  :  me,  where  n  and  m 
may  be  any  rational  numbers,  the  first  greater,  and  the 
second  either  greater  or  less,  than  unity. 

This  is  the  least  artificial  class  of  notation,  as  no  contrac- 
tions are  employed  ;  but  it  is  on  that  account  rather  incon- 
venient, the  symbols  being  somewhat  cumbrous  in  use.  It 
is  employed  in  the  works  of  the  Berlin  school  of  minera- 
logists, including  those  of  Weiss,  Rose,  and  Rammelsberg. 

Miller's  notation.  If  in  a  series  of  planes  like  those  of 
fig.  5,  the  unit  form  be  supposed  to  lie  outside  the  others, 
the  intercepts  of  any  enclosed  plane  upon  two  of  the  axes 
may  be  expressed  as  fraction  of  an  unit  intercept,  which 
then  becomes  the  longest  instead  of,  as  in  Weiss's  method, 
the  shortest.  Thus,  in  fig.  5,  supposing  a  plane  to  be  drawn 
through  A'  B'  c'  parallel  to  ABC,  the  lengths  of  the  intercepts 
will  all  be  multiplied  by  three,  which  does  not  alter  their 
ratios,  so  that  the  new  plane  will  still  have  the  symbol 
a  :  b  :  c  as  before  ;  but  the  relation  of  the  plane  A  B"  c'  to  the 

new    axes   is  -:-:-,  and  if  the  symbol  of  the  unit  face 
3     2     i 

be  written  as  -  I  -  :    ,  the  denominators   of  these  groups 
iii 

of  fractions  alone,  if  written  down  in  the  proper  order — thus, 
iii,  32  i — give  a  new  kind  of  symbol  perfectly  expressing 
the  axial  relations  of  the  two  planes.  This  method  is  adopted 
by  Miller,  the  numbers  making  up  the  symbols  being  called 
indices.  These  are  always  whole  numbers  including  o,  and 


CHAP.  II. ] 


Millers  Notation. 


in  the  greater  number  of  instances  not  exceeding  6.  Intercepts 
on  the  negative  side  of  any  axis  are  indicated  by  a  minus 
sign  written  above  the  corresponding  index  number.  Thus, 
the  counterpart  planes  of  1 1 1  and  321  are  1 1 1  and 


FIG.  6. 


«J 

The  general  expression  for  a  plane  intercepting  three 
axes  at  unequal  distances  in 
this  system  is  obtained  by  the 
method  shown  in  fig.  6,  where, 
supposing  H  K  L  to  represent 
such  a  plane,  and  ABC  that  of 
the  unit  form,  the  axial  lengths 
of  the  former  will  be  fractions 
of  the  latter,  as  in  the  pre- 
ceding figure,  or  putting 

i  i 

o  H  =  T  o  A,  OK  =  T  o  B,  and 


o  L  =  T  o  c,  will  be  represented  by 


OA 


OB 


OC 


o  H  =  -,  ,  o  K  =  -r-,  and  o  L  = 

and,  conversely,  the  indices  will  be 

OA         OB  oc 

h  =  —  ,  £  =  --  .and  /= — . 

OH'         OK'  OL 

These  values  are  in  no  wise  altered  by  multiplying  them  by 
any  numerical  co-efficient,  as  the  unit  parameter  will  be 
similarly  altered  by  the  same  process,  and  the  ratio  of  the 
two  series  will  be  unchanged.  The  symbol  /j^/is  therefore 
the  most  general  expression  that  can  be  obtained  for  a  form 
by  this  method  of  notation,  where  all  three  indices  are  dissi- 
milar ;  and  from  it  the  symbols  of  all  other  planes  in  a  system 
may  be  obtained  by  assigning  particular  values  to  the  different 
indices.  Thus,  i  i  i  is  a  special  form  in  which  h  •=  k  =  / ; 
o  o  r  another  where  h  =  k  =  o  and  /=  i ;  and  so  on  with  all 
possible  values  that  can  be  assigned  to  them.  This  system 
of  notation  by  indices  was  first  used  by  the  Rev.  Dr.  Whewell, 


2o  Systematic  Mineralogy.  [CHAP.  II. 

but  has  been  developed  and  extended  by  Professor  W.  H. 
Miller,  of  Cambridge,  to  whom  it  is  substantially  due  in  its 
present  form.  It  is  by  far  the  most  elegant  of  all  the  methods 
in  use,  and  will  probably  be  adopted  at  no  very  distant  date 
by  all  mineralogists,  although  up  to  the  present  time  its  use 
has  been  somewhat  restricted. 

If  Miller's  symbol  of  the  unit  plane  (m)  corresponding 

a     b     c 
to  Weiss's  a  :  If  :  c,  be  written  as  -  '  ~.v~-,  that  of  hkl  will 

be  7  :  -  :  ~,  which  ratios,  when  brought  to  whole  numbers 
hkl 

by  multiplying  each  term  by  hkl,  become  kla  :  hlb  :  hkc, 
or  the  parameter  co-efficient  of  any  axis  will  be  the  product 
of  the  indices  of  the  other  two,  and  these  products,  when 
written  in  the  proper  order  and  reduced  to  their  simplest 
expression,  will  give  the  required  symbol.  For  instance, 
Weiss's  symbol  corresponding  to  (hkl}  =  (i  2  3)  is 

a  .  b  ,  c 

1  '  2  '  z=        ''        ''  2f==3a  :  ?2     :  C- 

Similarly  (346)  corresponds  to 

a     b     c 

~  :  —  :  £  =  2Aa  :  iS<£  :  \2.c-=  2a  :  3Y>  :  c. 

346 

When  one  index  =  o,  the  corresponding  co-efficient 
=  ao,  and  the  co-efficients  of  the  other  two  axes  are  found 
by  interchanging  their  indices  ;  thus  (3  2  o)  =  2  a  :  3  <5>  :  oc  r 
=  a  :  f  b  :  oo  c.  When  two  indices  or  co-efficients  =  o 
or  QO,  the  third  index  or  co-efficient  is  put  =  i,  which  means 
that  the  intercept  on  that  axis  is  finite,  and  not  that  it  is  of 
any  particular  unit  length  ;  thus  (o  o  i)  =  GO  #  :  oo  b  :  c ; 
(o  i  o)  =  <x>  a  :  l>  :  cc  c ;  and  (ioo)  =  «:oo^:  GO  c. 

Weiss's  numbers  being  directly,  and  Miller's  inversely, 
proportional  to  the  lengths  of  the  intercepts,  the  highest 
co-efficient  will  always  correspond  to  the  lowest  index,  what- 
ever may  be  the  order  of  the  symbols.  When  h  is  the  highest 
and  /  the  lowest  of  three  dissimilar  indices  h  k  /,  the  corre- 


•CHAP.  II.]  Conversion  of  Symbols.  21 

spending  ratio  of  the  co-efficients  in  the  same  order  will  be 
i  :  n  :  m  where  n  >  i  and  m  >  n.     Dividing  this  by  m  n  it 

becomes          :   ~  :  ~,  whence  are  deduced  h  =  ;// ;/,  k  =  m. 

m  n     m     H' 

and  /  =  «,  as  the  expressions  for  the  conversion  of  Weiss's 
into  Miller's  symbols. 
Thus  : 

a  :  -it-  a  :  4  a  —  (V  4  -3)  =  (l6  I2  4)  =  (4 3  *)• 

The  above  case  applies  to  the  cubic  system,  where  the 
positions  of  the  signs  are  interchangeable  ;  in  the  remaining 
systems  n  is  always  >  i,  but ;//  is  specially  restricted  to  the 
vertical  axis,  and  may  be  either  greater  or  less  than  n  or  i. 

Thus  : 
a  :  |  b  :  ^-  c  =  (3  2  6) ;  i! a  :  b  :  2  c  =  (4  6  3),  &c. 

In  arranging  the  indices  or  other  signs  forming  the 
symbol  of  any  face,  care  must  be  taken  that  the  axes  are 
always  noted  in  the  same  order.  Unfortunately,  there  is  no 
uniformity  of  practice  in  this  matter,  either  one  of  the  three 
axes  being  considered  as  the  first  by  different  authors.  In 
Miller's  system  the  three  axes  are  indicated  by  the  letters 
x  y  z,  the  first  extending  right  and  left,  the  second  from  front 
to  back,  and  the  third  above  and  below  the  centre.  Weiss 
calls  the  axes  a  b  c,  their  order  being  a  front  and  back,  b 
right  and  left,  and  c  top  and  bottom.  The  latter  system,  as 
being  the  oldest  and  most  generally  known,  is  adopted  in 
this  volume.  The  position  of  the  first  and  second  indices 
in  the  symbol  of  any  face  as  noted  in  the  figures  must  there- 
fore be  transposed  to  make  them  correspond  with  those  of 
Miller's  order.1 

Naumamis  notation.  The  symbol  of  any  face  may  be 
used  by  implication  to  indicate  the  whole  of  the  same  kind 
of  faces  in  a  crystal,  if  the  symmetry  of  the  system  is  known. 

1  The  difference  expressed  in  words  is  as  follows  :  The  plane  whose 
parameters  are  positive  on  all'  three  axes,  is,  according  to  Miller,  the 
right  front  top  face,  while  in  Weiss's  order  it  is  the  front  right  top  one. 


22  Systematic  Mineralogy.  [CHAP.  n. 

This  is  apparent  in  Weiss's  notation  by  the  use  of  index 
letters  for  the  different  axes  ;  but  not  in  Miller's,  where  the 
symbols  are  similar  for  all  the  systems,  and  do  not  of  them- 
selves show  the  character  of  the  form.  Another  method, 
modified  from  that  of  Weiss,  due  to  the  late  Dr.  C.  F. 
Naumann,  indicates  both  form  and  symmetry  in  a  single 
symbol  by  combining  the  parameter  values  with  certain 
arbitrary  signs,  that  differ  in  each  system,  representing  the 
form.  In  the  cubic  system,  where  the  three  axes  are  all 
rectangular  and  the  parameters  equal,  the  unit  form — 
Miller's  1 1 1  or  Weiss's  a  :  a  :  a — is  the  regular  octahedron. 
This  is  represented  by  the  capital  letter  O,  the  initial  of  octa- 
hedron. In  the  other  systems,  where  the  forms  correspond- 
ing to  the  unit  values  of  the  parameters  are  of  the  kind 
known  as  pyramids,  the  initial  P  is  used  as  the  unit  symbol. 
The  derived  forms  in  any  system  are  indicated  by  the  addition 
of  Weiss's  co-efficients,  according  to  the  number  of  axes  on 
which  the  intercepts  vary  from  unity,  but  the  unit  letter  is 
never  used  more  than  once  in  any  symbol.  The  general 
symbol  corresponding  to  hk  /,  or ;//  a  :  b  :  n  c,  is  m  P  n  in  the 
systems  with  rectangular  axes,  but  in  those  with  one  or 
more  oblique  axes  accents  and  other  arbitrary  signs  are 
added  to  the  characteristic  letter  to  show  the  symmetry  of 
the  system.  This  notation  is  more  extensively  used  than 
any  other,  mainly  from  the  circumstance  that  the  most 
popular  text-book  on  Mineralogy  is  written  by  its  author  j  * 
and  in  a  slightly  modified  form  it  is  also  used  in  Dana's 
manual  and  text-books,  which  are  probably  the  most 
abundantly  circulated  books  of  their  class  in  the  English 
language.  The  symbols  have  the  advantage  of  being  short 
and  convenient  ;  and  being  essentially  arbitrary,  when  their 
nature  is  once  understood  they  cannot  be  mistaken  for  any- 
thing else,  and  are  therefore  well  suited  for  descriptive  pur- 

1  Elemcnte  dcr  Mincralogie.  C.  F.  Naumann.  The  first  edition 
was  published  in  1846,  and  the  ninth  sHortly  before  the  author's  death 
in  1873.  A  tenth  edition,  edited  by  F.  Zirkel,  appeared  in  1877.  The 
same  system  is  followed  in  the  late  Professor  NicoFs  manual. 


CHAI-.  II.]  Levy's  Notation.  23 

poses.  They  are,  however,  only  adapted  for  indicating 
forms — that  is,  the  whole  system  of  faces  corresponding  to 
any  particular  set  of  parameters,  and  not  individual  planes 
—as  the  parts  of  the  symbols  are  arranged  in  an  invariable 
and  arbitrary  form.  The  special  method  of  arranging  these 
for  different  kinds  of  symmetry  will  be  considered  under  the 
description  of  the  different  systems. 

Levy's  notation.  In  a  system  with  three  axes  the  solid,  con- 
tained by  faces,  parallel  to  two  of  the  axes,  and  intersecting 
the  third  at  some  measurable  distance,  will  either  be  a  cube 
or  some  parallelepipedon,  and  in  the  hexagonal  system  it 
will  be  an  hexagonal  prism.  In  such  forms,  when  exactly 
developed,  the  edges  will  be  of  the  same  length  as  the  axes  to 
which  they  are  parallel,  and  the  plane  angles  of  the  parallelo- 
grams forming  the  faces  will  have  the  characteristic  angles  of 
the  axes.  The  system  originally  invented  by  the  Abbe  Haiiy, 
and  subsequently  modified  by  Levy  and  Descloizeaux,  uses 
reference  solids  of  this  kind  known  as  '  primitive '  forms, 
which  are  essentially  the  same  as  the  molecular  networks  of 
Bravais.  The  faces  in  such  forms  are  indicated  by  the 
capital  letters  P  M  T,  their  points  by  vowels  a  e  i  o,  and 
their  edges  by  consonants  in  an  invariable  order,  from  left 
to  right,  each  primitive  form  requiring  as  many  letters 
for  its  description  as  it  has  different  parts.  The  num- 
ber of  these,  therefore,  is  an  indication  of  the  degree  of 
symmetry.  The  symbols  of  derived  faces  are  compounded 
of  those  of  the  part  of  the  primitive  modified,  whether  an  edge 
or  a  solid  angle,  and  a  series  of  signs  indicating  the  ratio  of 
the  intercepts  of  the  new  plane  upon  the  edges,  written  as 
exponents.  Thus,'  in  the  cubic  system,  the  octahedron  is 
written  as  rt1,  which  means  that  it  intercepts  an  equal  length 
upon  each  of  the  edges  measuring  from  point  a  where  three 
faces  meet.  This  is  the  oldest  of  all  the  systems  of  nota- 
tion, and  was  at  one  time  almost  universally  current,  but  at 
present  it  may  be  considered  as  restricted  to  the  mineralogists 
of  France,  by  whom  it  is  generally  used.  The  important 
works  of  Descloizeaux  and  Mallard  being  written  in  this 


Systematic  Mineralogy. 


[Cii.\r.  II. 


system,  a  knowledge  of  its  principles  will  be  found  useful  to 
the  student.1 

Relations  of  faces.  The  symbol  of  any  face  in  a  crystal 
may,  by  an  extension  of  meaning  be  considered  as  typical 
of  the  whole  form — that  is,  of  all  faces  similarly  related  to  the 
axes  of  reference.  When  this  is  meant,  the  symbol  is  en- 
closed in  brackets,  thus —  {/i  k  1} ;  but  when  restricted  to  an 
individual  face  it  is  put  in  parentheses,  thus — (///&/).  As 
will  be  subsequently  seen,  the  possible  number  of  faces  in  a 
form  varies  with  the  symmetry  of  the  system,  the  maximum 
of  forty-eight  occurring  in  the  cubic,  and  the  minimum  of 
two — that  is  the  face  and  its  counterpart — in  the  triclinic 
system.  There  are  many  instances  of  crystals  contained 
by  the  faces  of  only  a  single  form,  especially  in  the  cubic 
system ;  but  it  is  far  more  common  to  find  them  made 
up  of  two  or  more  forms  grouped  in  regular  order,  such 
compound  crystals  being  known  as  combinations.  There 
is  no  limit  to  the  number  of  forms  that  may  enter  into 
a  combination,  subject  to  the  conditions  that  they  all 
have  the  same  degree  of  symmetry,  and  are  so  arranged 

that  all  the  faces  meet  in 
convex  angles.  Crystals,  in 
which  some  of  the  faces  meet 
in  concave  or  re-entering 
angles,  are  not  uncommon, 
but  these  are  never  simple, 
being  peculiarly  arranged 
groups  of  two  or  more, 
known  as  twin  crystals. 
The  general  solution  of  the 
problem  of  the  determina- 
tion of  the  direction  of  the 

edge  or  line  intersection  of  two  dissimilar  faces  in   terms 
of  their  parameters  is  as  follows  : — Let  H  K  L,  H'  K'  L'  (fig.  7) 

1  A  good  account  of  it  will  he  found  in    Pisani's  Traitc  dc 
ralogie. 


FIG.  7 


CHAP.  II.]  Relations  of  Faces.  25 

be  two  such  planes,  their  intersections  H  K  and  H'  K',  with  the 
axial  plane,  A  o  B,  cross  in  the  point  y,  which  will  therefore 
be  a  point  in  the  required  edge  as  common  to  both  faces  ; 
ft  will  be  a  second  point  of  a  similar  kind  in  the  plane 
A  o  c,  and  a  a  third  in  the  plane  A  o  B.1  As  the  direction  of 
the  line  of  intersection  is  unchanged  by  moving  either  face 
parallel  to  itself,  the  face  (//  k'  /')  if  shifted  to  the  position 
ELF  gives  the  new  line  L  D  as  the  required  direction.  This, 
however,  is  equivalent  to  multiplying  the  parameters  of  the 

face   by  — 7.     These  latter  will  therefore  be  for  the  new 
'  OL' 

position 

o  K  =  o  H'  P-^-,  o  F  =  o  K'  ^,  and  o  L 
o  i/  OL' 

— that  is,  the  intercept  on  the  third  axis,  common  to  both 
faces,  is  an  original  parameter  of  (h  k  1}  ;  the  position  of  the 
point  L  is  therefore  determined.  To  find  the  point  D, 
draw  in  fig.  8,  D  u  parallel  to  o  B,  and  D  v  parallel  to  o  A, 
when  the  problem  takes  the  form  of  the  determination  of 
the  co-ordinates  of  the  point  D,  the  lengths  o  u  and  o  v,  in 
terms  of  the  parameters,  as  when  these  are  known,  the  sides 
of  the  parallelogram  o  u  D  v,  and  with  them  the  position 
of  D,  are  determined. 

The  triangle  o  K  H   is  similar    to   u  D  H,  also   o  F  E   is 
similar  to  u  D  E,  whence  follows 

OK  :  UD=OH  :  UH  =  OH  :  (OH  —  o  u) 
o  F  :  u  D  =  o  E  :  u  E  =  o  E  :  (o  E  —  o  u). 

The  first  of  these  ratios  gives  the  equation 

OK  .  OH  — OU.  OK  =  O  H  .  U  D, 

and  the  second, 

OF.OE  —  OU.OF  =  OE.UDj 

1  This  demonstration  is  given  in  Groth's  Physikalische  Krystallo- 
graphie. 


26  Systematic  Mineralogy.  [CHAP,  ir.. 

whence  the  two  unknown  quantities  o  u  and  u  D  are  derived 
as  follows  : 

OE.OK.OH— OF.OE.OH 
OE.  OK  — O  H  .  O  F 

OK. OF. OE  — OK.  OH. OF 
U  D  =  — 

OE.  OK  — OH  .O  F 

or, 

OK  — OF 


OU=  OH.  OE 
=  OV=OK.OF 


O  E.  OK  — O  F.  O  H 

OE  — OH 
OE.  O  K  — OF.  OH' 


Substituting  the  proper  values  for  o  E  and  o  F,  these  become 


OK-°i..OK' 
OL 


O  L        O  L      .         O  L 

. .  0  H'  .  O  K  —  ..  O  K'  .  O  H 

O  L  O  L 

OL' 
-.OK-OK' 

=  OH.°-I1.0H'.0_L 


OL''  O  H'.  OK  — OK'.  OH 


OL      , 

— , .  OH'  — OH 


o  v  =  o  K  .  —  .OK' 


OL  OL 

—  .OH   .OK  ---    .OK   .OH 
O  L'  O  L' 


OH'——.  OH 
=  O  K  .  ,  .  O  K'  .         °  L 


ou  = 
o  v  = 


OH'. OK  —  OK'. OH 

OH  .  OH'  OK.OL'— OL.OK' 

OL'  OK.  OH'— OH.  OK' 

o  K  .  o  K'  OL.OH'  —  OH.OL' 

OL'  •  OK.  OH'— OH.  OK' 


— which  are  the  equations  for  the  co-ordinates  of  the  point  D 
in  terms  of  the  parameters. 


CHAK  II.] 


Relations  of  Faces. 


FIG.  8. 


If  in  fig.  8  a  length  o  w  =  o  L  be  laid  off  on  the  negative 
side  of  the  axis  c,  and  the  parallelepiped  o  u  D  v  w  R  Q  s  con- 
structed, the  diagonal  o  Q  will 
also  give  the  direction  of  the 
edge  between  the  two  faces 
being  parallel  to  LD.  The 
multiplication  of  all  the  para- 
meters of  (//  k  /)  by  any  quan- 
tity, »i,  does  not  affect  the 
direction  of  the  plane  or  of  its 
intersection  with  the  other 
plane,  so  that,  if  for  OH,  OK, 
o  L  in  the  preceding  formulas 
the  new  values  m  o  H,  m  o  K, 
and  m  o  L  be  substituted,  we 
obtain  o  v'  =  ;//  o  v,  o  v'  = 
m  o  v,  o  w  =  m  o  w — that  is, 
all  the  sides  of  the  above  parallelepiped  will  be  multiplied  by 
the  same  quantity,  whereby  the  'direction  of  its  diagonal  is 
not  altered.  The  same  holds  good  when  the  expressions 
for  o  u,  o  v,  and  o  w  are  multiplied  by 

o  K  .  OH'  — OH.  OK' 


-  s 


O  H  .  O  H'.  O  K  .  O  K'.  O  L' 

whence   the    following    perfectly   symmetrical    expressions 
result  : 

i  i 


o  u  =  - 
o  v  = 
ow  = 


OL.  OK'  OK.  OL' 
i  i 

OH.OL'  ~~  OL.OH'' 
i  i 


OK.  OH'      OH.  OK' 

In  the  application  of  these  equations  to  the  determina- 
tion of  the  direction  of  intersection  of  two  faces,  this  inter- 
section is  supposed  to  pass  through  the  origin  of  the  axes. 


28  Systematic  Mineralogy.  [CHAP.  II- 

which  gives  one  point ;  the  second  is  found  by  laying  off 
upon  the  three  axes  the  values  found  for  the  parameters,  in 
the  proper  directions,  positive  or  negative,  according  to 
their  signs,  which  gives  the  three  sides  of  the  parallelepiped 
whose  diagonal  drawn  from  the  origin  of  the  axes  is  the 
-direction  required. 

If  it  is  required  to  express  o  u,  o  v,  o  w,  by  the  indices 
instead  of  the  parameters  of  the  faces,  the  values  of  the 
latter,  expressed  by  the  former,  must  be  introduced  into  the 
equation,  or  for  the  unit  parameters  a,  b,  c, 

a  b  c 

~  li                    ~  V                   ~  I' 
, a  i b  f c 

~  Ji"  ~  F  =  /'' 

If  these  values  are  substituted  for  o  u,  o  v,  o  w,  in  the 
formulae  given  above,  and  each  expression  is  brought  to  one 
denomination,  we  obtain — 

kl'-lk'       u 


o  u  = 
o  v  = 
ow  = 


be  bt 

lh' -hi1  _  7- 

a  c  etc 

hk>-kh'  _w 

ab       ~  atf 


when  the  differences  forming  the  numerators  are  expressed 
by  the  contractions  u,  z>,  w.  Multiplying  these  by  the  pro- 
duct a  b  c,  which  does  not  alter  their  relative  magnitudes, 
they  become 

a  u,  b  v,  c  w, 

where  the  sides  of  the  parallelepiped  whose  diagonal  is  the 
required  direction  are  represented  by  magnitudes  depending 
only  on  the  indices.     For  their  determination  the  following 
method  is  given  by  Miller,  which  is  easily  remembered  : — 
Write  down  the  indices  one  above  the  other  twice,  cut 


CHAP.  II.]  Principle  of  Zones.  29 

off  the  first  and  last  columns,  multiply  the  others  crosswise,. 
and  subtract  one  product  from  the  other.     Thus  : 

h\k      I      h      k\  I 
h'    K  x  f  x  h' x  k' !  /' 


kl-lK,lh'  -hl',hk'-kh' 

•=11         =7'  •=  W, 

which,  like  the  indices,  are  rational  whole  numbers. 

Zones.  Any  number  of  faces  parallel  to  any  right  line  is 
called  a  zone.  Faces  belonging  to  the  same  zone,  and 
which  consequently  intersect  in  lines  which  are  parallel  to- 
one  another  and  to  every  face  of  the  zone,  are  said  to  be 
tautozonal.  If  we  suppose  the  faces  of  a  zone  to  pass  all 
through  one  point,  they  will  intersect  in  a  right  line  passing 
through  that  point,  and  the  direction  of  that  line  is  called 
the  zone  axis.  For  instance,  in  the  cube,  the  front  and  back 
and  right  and  left  pairs  of  faces  constitute  a  zone  whose 
edges  are  all  vertical  and  parallel  to  the  axis  c,  which 
is  therefore  their  zone  axis  ;  and  either  of  these  may  be 
grouped  with  the  third  pair  of  faces,  forming  zones  whose 
axes  are  a  and  b  respectively. 

Any  three  faces,  P  Q  R,  whose  symbols  are  (efg)  (h  k  /) 
(P  y  r\  w'll  ue  m  the  same  zone  when  the  line  of  intersec- 
tion of  P  and  Q  is  parallel  to  that  of  Q  and  R,  or,  in  other 
words,  when  the  diagonals  of  the  parallelepipeds  found  by 
the  method  shown  in  fig.  8  coincide  for  each  pair.  Thisr 
however,  is  only  possible  when  the  ratios  of  the  three  sides 
are  the  same  in  both  —  that  is,  when  the  sides  of  the  first  only 
differ  from  those  of  the  second  by  a  constant  factor.  The 
intersection  of  P  and  Q  is  determined  by  the  parallelepiped, 
whose  sides  are  — 

an  =  a  (fl  —  gk\  b  v  =  b  (gh  —  el\  c  w  -=c(ek  —fh\ 


and  that  of  Q  and  R  by 
a  n'  =  a  (k  r  —  I  q\  b  v'  =  (I  p  —  h  r),  c  w'  =  c'  (h  q  —kp). 

If  the  faces  p  Q  R  are  tautozonal,  their  sides  must  be  to 


3O  Systematic  Mineralogy.  [CHAP.  II. 

each  other  in  the  ratio  of  a  common  constant,  or,  calling 
the  latter  c  — 


kr-lq  (i) 

c  (gh-  el)  =  lp  -  hr  (2) 

c  (ek-fh}  =  hq-kp  (3) 


in  which  equations  the  axial  lengths  are  eliminated  by  ap- 
pearing on  both  sides  as  factors.  Multiplying  (i)  by  <?,  (2) 
by/  and  (3)  by  g,  and  adding  all  three  together,  the  result- 
ing equation  becomes 

ekr  —  elq  -\-flp  —fh  r  +  gh  q  —  gkp  =  o,1 

which  is  the  condition  of  tautozonality  for  the  three  faces, 
p  Q  R.  If,  as  in  the  previous  case,  we  make 

//  —  g  k  =  u,  gh  —  c  1=  v,  ek  —fh  =  ov, 

the  last  equation  becomes 

up  +  v  q  +  w  r  =  o, 

which  shows  that  the  condition  required  to  bring  three 
planes  into  the  same  zone  depends  only  on  their  indices, 
and  is  completely  independent  of  the  lengths  of  their  axes. 

The  quantities  u  v  w  are  called  the  indices  of  the  zone, 
and  to  distinguish  them  from  those  of  a  face  or  form  they 
are  written  in  square  brackets,  thus  \u  v  w],  forming  the  so- 
called  zone  symbol  ;  the  face,  if  any,  to  which  this  symbol 
belongs  is  called  the  zone  plane,  and  is  perpendicular  to  the 

1  This  condition  may  be  obtained  directly  from  the  consideration 
that  three  planes  through  the  origin,  parallel  to  the  three  faces,  have 
for  their  equations,  expressed  as  in  ordinary  geometry, 


a  b 

and  the  condition  that  the  three  planes  should  be  parallel  to  a  line  is 

obtained  by  eliminating    -,    -^  ,   and  —  from  the  three  equations.  This 
a      l>  c 

gives  the  determinant  above  written. 


CHAP.  II.]       Determination  of  Faces  by  Zones.  31 

.zone  axis.  It  may  be  determined  from  the  intersection  of 
any  two  planes,  P  and  Q,  out  of  those  forming  the  zone,  and 
from  it  the  whole  number  of  possible  tautozonal  faces  may 
be  calculated  by  substituting  for  q  and  r  successively  all  the 
•simple  rational  numbers  o,  i,  2,  .  .  .  and  calculating,  in 
accordance  with  the  above  condition,  the  corresponding 
values  of/. 

Determination  of  a  face  by  two  zones.  As  a  plane  is 
determined  when  the  positions  of  two  straight  lines  parallel 
to  it  are  given,  that  of  the  face  of  a  crystal  lying  in  two 
zones,  and  therefore  parallel  to  the  axes  of  both,  is  simi- 
larly determinable.  If  the  symbols  of  the  zones  are 

[//  v  w~\  and  \u'  v'  w'\, 

the  indices  /  q  r  of  the  face  must  satisfy  the  equation  of 
condition  in  regard  to  both.  Consequently 

n  p  +  v  q  +  iv  r  =  o, 
u' p  +  v'  q  +  w'  r  =  o, 

whence  we  derive 

v  w'  —  w  v' 


p  =  r 
q  —  r 


UV   —V  U 

wu'  — 11  w' 
u  v'  —  v  u' 


One  of  the  three  indices  may,  however,  be  made  equal  to 
any  number  at  pleasure  ;  say,  for  example, 

r-=u  v'  —  v  if', 
which  gives 

p  =  71  w'  —  w  v', 
q  •=  w  u'  —  u  w', 

as  the  three  indices  of  the  face  common  to  both  zones. 
These  indices  may  be  derived  from  those  of  the  zones  by 
the  scheme  of  cross  multiplication  and  subtraction,  in  the 


Systematic  Mineralogy. 


[CHAP.  II.. 


same  way  that  those  of  a  zone  are  found  from  those  of  two 
of  its  faces.     Thus  : 


v   U  v 

' X     ' X 


w 
w' 


ww'  —  v'  W,  wu'  —  u  w,'  u  v'  —  vu' 

=p      =9      —  r- 

By  this  method  the  symbols  of  a  face  may  be  found' 
when  those  of  any  two  zones  in  which  it  lies,  or  of  any  two 
faces  in  each  of  those  zones,  are  known.  Suppose,  for 
example,  a  face  is  observed  to  lie  in  one  zone  with  the  faces 
(123)  and  (i  13),  and  in  another  with  (o  i  i)  and  (i  2  z}T 
the  corresponding  zone  symbols  will  be 


xxx 


-3>  3~3>  i-2=[3°  J]        2~2>   i-o,  o-i=[c  i  i]; 
and  that  of  the  face  common  to  both 


3 

0 

0 

I 

X 

1 
I 

X 

3 

0 

X 

0 

I 

o-i,  0-3,  3-0  =  (i  33). 

Any  three  faces  intersecting  at  any  angles  may  appear  in 
the  same  zone,  but  a  fourth  (or  any  further  number  of  faces)T 
arbitrarily  placed,  is  not  possible  according  to  the  principle 
of  rationality,  or  rather  according  to  the  particular  develop- 
ment of  it  known  as  the  principle  of  anharmonic  ratios. 
This  principle  is  of  such  importance  that  a  general  demon- 
stration of  it,  obligingly  furnished  by  the  editor,  Mr. 
Merrifield,  is  given  in  the  following  pages. 


ANHARMONIC  RATIOS. 

Let  A  B  c  D  (fig.  9)  be  any  four  points  on  a  right  liner 
and  let  o  be  an  arbitrary  point  through  which  the  right 


CHAP.  II.]       Anharnwnic  Ratios  of  Planes. 


33 


FIG.  9. 


lines  o  A,  o  B,  o  c,  o  D  are  drawn.     Also,  let  fall  o  P  perpen- 
dicular on  A  D.     Then  we  have 


OB  =  AB.OP)  "j    Since  each  side  of  the  equa- 
OD  =  BD.OP)    I    tion  is  a  different  expres- 
sion for  twice  the  area  of 
a  triangle. 


O  A  .  O  B  Sin  A 
O  B  .  O  D  Sill 

o  A  .  o  c  sin  A  o  c  =  A  c .  o  P  I 
o  c .  o  D  sin  COD  =  CD.OPJ 

Dividing  out  suitably — 

sin  A  o  B  .  sin  A  o  c 
sin  BOD*  sin  COD 


A  B 
B  D 


A  c 
c  D' 


which  is  the  anharmonic  ratio  of  the  four  points  or  four  lines  : 
the  second  side  of  this  equation  shows  that  the  ratio  is  inde- 
pendent of  the  position  of  the  point  o,  which  may  therefore 
be  at  o'.  The  first  side  shows  that,  the  pencil  being  given, 
the  transversal  is  immaterial.  Thus  the  anharmonic  ratios 


A  B 
BD 


AC 
CD' 


AB 


A0CC 


A'B'   .  A/C' 

iV  :  cV 


are  all  equal,  and  this  whether  OAD,  O'AD  are  in  the  same 
plane  or  not. 

There  is  more  than  one  anharmonic  ratio  of  four  points 
on  a  line.     They  are  connected  by  means  of  the  identity, 


34  Systematic  Mineralogy.  [CHAP.  n. 

which  is  obtained  by  the  cyclic  permutation  of  the  letters 
E  c  D  in  the  expression  A  B  .  c  D. 

The  ratio  of  any  two  of  the  terms  in  this  formula  may 
be  taken  as  the  anharmonic  ratio  of  the  pencil. 

A  further  consequence  of  this  property  is,  that  if  we  con- 
sider o  and  o'  as  in  separate  planes,  OO'A,  OO'B,  oo'c,  OO'D 
constitute  a  pencil  of  planes  such  that  the  anharmonic  ratio 
of  any  transversal  line  is  invariable,  and  equal  to  the  corre- 
sponding sine-ratio  of  the  dihedral  angles.  If  any  four 
points  on  a  line  are  given,  any  pencil  of  planes  through  the 
four  points  will  have  the  anharmonic  ratio  of  the  range,  and 
every  transversal  of  any  such  pencil  will  again  have  the 
same  anharmonic  ratio. 

The  sine-identity  of  a  pencil  is  easily  written  down 
from  the  corresponding  one  of  a  range  by  simply  inserting 
the  letter  at  the  vertex,  and  the  word  sine.  Thus — 

AB.CD  +  AC.DB  +  AD.BC  =  O,  gives 

sin  AOB.sincoD  +  sin  Aoc.sin  DOB  +  sinAOD.sin  BOC  =  O, 
and  the  anharmonic  sine-ratios  are 

sin  A  o  B  .  sin  c  o  D  :  sin  A  o  c .  sin  D  o  B,  &c. 

N.B. — Points  on  a  right  line  are  a  range.  Right  lines  meeting  in 
a  point  are  &  pencil.  Planes  meeting  in  a  right  line  are  also  a  pencil. 

ANHARMONIC  PROPERTY  OF  ZONES. 

The  anharmonic  property  of  zones  is  this  :  that  if  we 
take  in  space  four  planes  parallel  to  four  of  the  faces  of  a 
zone,  and  meeting  in  one  line,  the  anharmcnic  ratio  of  the 
pencil  is  rational. 

It  is  to  be  remembered  that  all  the  faces  of  a  zone  are 
parallel  to  a  line.  Hence  all  their  intersections,  two  and  two, 
will  be  in  parallel  lines.  The  necessary  construction  is 
therefore  at  once  obtained  by  drawing,  through  the  inter- 
section of  any  two  faces,  planes  parallel  to  the  other  two. 

The  four  faces  of  the  zone,  by  the  ordinary  law  of  crystal- 


CHAP.  II.]        Anharmonic  Ratios  of  Zones. 


35 


lography,  cut  off  from  each  of  the  axes  intercepts  having 
rational  numerical  ratios;  and  what  we  have  to  prove  is  that, 
as  a  consequence  of  this  rationality,  the  corresponding  an- 
harmonic  ratio  is  also  rational. 

Confining  our  attention  to  one  plane  only,  let  us  con- 
sider four  transversal  lines  meeting  the  two  axes  :  thus,  in 
the  annexed  figure, 

fl^j,  #2^2>  a3^3'  a^^i  are  f°ur  transversals  to  ox  and  07, 
so  taken  that 


Through  £,  draw  b^a]  '2  parallel  to 


We  want  to  show  that  if  Imn pqr  are  rational  the  anhar- 
nionic  ratio  of  the  pencil  &l  aY  a'2  a'3  a\  is  also  rational. 
This  ratio  is 


_  R 


Now 


fm    ^ 

=  (  ~-i  ) 


36  Systematic  Mineralogy.  [CHAP.  II. 

i        ;  /  /  fn 

a2a'4  =  oa'4-oa  '2  -     -  — 
\r 


oa, 
n 


_ 

m  n     m      (rn—q)(np—li-) 

q~         r~  q 


which  is  rational,  \ilmn  pqrzre  so. 

The  whole  of  the  anharmonic  ratios  of  the  zone  are  con- 
tained in  the  ratios  two  by  two  of  the  terms  in  the  identity  — 


Since  Imn  pqr  are  either  integral,  or  else  rational  numerical 
fractions,  each  term  of  this  identity  must  be  so  too,  and  the 
ratio  of  any  two  terms  must  be  a  rational  fraction. 
As  a  numerical  example,  let 

/=2>  ;w  =  3,  «  =  4,^=  3,  ?  =  5,  r=8; 
then  the  above-written  identity  becomes 


and  the  anharmonic  ratios  are  any  one  of  the  following  : 


In  practice,  the  dihedral  angles  will  be  measured,  and 
the  anharmonic  ratio  to  be  used  will  be  the  sine-ratio.  The 
test  of  whether  we  have  really  got  four  faces  of  one  crystal 
is  the  rationality  of  this  anharmonic  sine-ratio,  when  re- 
duced to  numbers.  It  will  usually  be  a  simple  numerical 
fraction,  in  low  terms. 


CHAP.  III.] 


Symmetry  of  Cube. 


37 


CHAPTER    III. 

CUBIC   SYSTEM.1 

THE  forms  of  this  system,  whose  symmetry  is  completely 
exhibited  in  an  actual  cube,  are  referred  to  three  principal 
axes  at  right  angles  to  each  other,  whose  unit  lengths  or 


FIG.  ii. 


FIG.  12. 


parameters  are  all  equal.  These  are  axes  of  quaternary 
symmetry,  each  one  passing  through  the  centre,  o,  and  joining 
the  centres  of  opposite  faces  perpendicularly  :  A  o  A,  BOB, 
c  o  c  (fig.  u).  They  lie  two  by  two  in  planes  parallel  to 
the  faces  which  are  the  principal  planes  of  symmetry. 

Next  in  order  are  the  four  axes  of  ternary  symmetry — ox', 
ox",  OT'",  ox""  (fig.  12),  making  with  one  another  angles 
of  109°  28'  16".  These  are  the  four  diagonals  of  the  cube, 
and  have  no  planes  of  symmetry  corresponding  to  them. 

Lastly,  there  are  six  axes  of  binary  symmetry,  the  lines 
joining  the  middle  points  of  opposite  edges — i;1  to  B(J 
(fig.  13).  They  lie  by  pairs  in  the  principal  planes  at  right 
angles  to  each  other,  at  45°  to  the  principal  axes,  and  are 


1  Other    names   are  Tesseral,   Regular,   Monometric,   and  Terqua- 
ternary. 


FIG.  13. 
B* 


B.-i 


38  Systematic  Mineralogy.  [CHAP.  ill. 

normals  to  six  corresponding  planes  of  symmetry.  These 
latter,  which  intersect  the  principal  planes  singly,  at  right 
angles  in  the  binary  axes,  by  pairs 
at  45°  and  135°  in  the  principal 
axes,  and  each  other  by  threes  in 
the  ternary  axes,  are  the  rectan- 
gular sections  of  the  cube  whose 

O 

sides  are  the  edges  and  face  dia- 
gonals, or  the  principal  and  binary 
axes,  and  whose  diagonals  are  the 
lines  joining  opposite  solid  angles, 
or  the  ternary  axes. 

The  symmetry  of  the  cube  is  not  altered  by  permuting 
the  order  of  the  principal  axes,  or  by  any  inversion  of  sign, 
and  the  preservation  of  these  symmetries  involves  the  re- 
tention of  all  the  others.  Taking  any  face  (hkt),  or 
(a  :  na  :  ma),  the  first  condition  requires  that  there  should 
be  six  similar  ones  due  to  permutation  of  indices,  or  co- 
efficients, namely — 


or 


hkl,  klh,lhk,  Ikh,  khl,  hlk, 
i  n  m,  n  m  i,  mm,  mm,  mm,  i  m  n. 


The  second  requires  that  for  each  of  these  combinations 
of  letters  there  should  be  faces  representing  each  of  the 
following  eight  permutations  of  signs  : 


Combining  all  these  together  we  find  that  to  satisfy  full 
cubic  symmetry  in  the  most  general  form,  any  face,  (hkl),  re- 
quires to  be  associated  with  47  others,  constituting  the  so- 
called  hexakisoctahedron.  In  its  most  regular  development, 
with  all  the  faces  equally  distant  from  the  centre,  it  is  con- 
tained by  48  plane  scalene  triangles  whose  edges  all  lie  in 


CHAP.  III. 


Hexakisoctahcdron. 


39 


FIG.  14. 


planes  of  symmetry.  These  edges  are  all  three  kinds  distin- 
guished in  fig.  14  as  long,  medium, 
and  short  (L,  M,  and  s).  Those  of 
medium  length  all  lie  in  the  prin- 
cipal planes  of  symmetry,  so  that 
the  sections  of  these  planes  are 
equilateral  but  not  regular  octa- 
gons, while  the  longer  and  shorter 
"ones  are  arranged  in  alternate 
pairs  in  the  other  six  planes  of 
symmetry,  the  sections  of  the 
latter,  therefore,  are  unequal 
eight-sided  figures.  The  dihe- 
dral angle  between  adjacent  faces,  or  the  so-called  interfacial 
angles,  are  always  greater  than  90°  and  less  than  iSo0,1 
They  are  for  the  particular  forms — 


{3  2  i}. 
{421}. 


162° 


M.  S. 

149°  oo,     158°  13', 
1 54°  47',    144°  03'. 


The  points  or  solid  angles  are  of  three  kinds — namely,  six 
eight-faced,  or  formed  by  the  meeting  of  four  long  and  four 
medium  edges  in  the  poles  of  the  principal  axes  ;  eight  six- 
faced,  formed  by  three  long  and  three  short  edges  in  each 
of  the  axes  of  ternary  symmetry,  which  are  also  known  as 
trigonal  interaxes;  and  twelve  four- faced,  formed  by  two 
short  and  two  medium  edges  meeting  in  each  of  the  axes  of 
binary  symmetry  or  rhombic  interaxes. 

Naumann's  symbol  for  this  form  is  mOn,  signifying  that 
two  parameters  vary  from  the  unit  length,  one  being  m  and 
the  other  n  times  greater  than  that  of  the  third  axis. 

The  geometrical  construction  of  the  octant,  including  the 

1  That  is,  no  face  can  be  at  right  angles  to  a  plane  of  symmetry,  or 
the  edges  of  symmetry  are  all  effective  edges  of  form. 


40 


Systematic  Mineralogy. 


[CHAP.  III. 


six  faces  whose  indices  are  all  positive  of  the  form  (3  2  i],  is 
given  in  fig.  I5.1 


FIG.  15. 


The  arrangement  of  the  symbols  of  the  faces  for  all  the 
positive  values  of  the  first  parameter  in  the  form  {421}  is 


FIG.  16. 


shown  in  fig.  19,  and  for  those  of  the  whole  of  the  general  form 
[hkl]  in  the  scheme  fig.  16.    This  supposes  the  edges  to  be 

1  In  this,  and  figs.  18,  21,  25,  and  28,  all  the  parts  are  similarly 
lettered  to  avoid  detailed  description. 


CHAP.  III.]  Icositetrahedron.  41 

projected  on  the  surface  of  an  octahedron,  which  is  then  flat- 
tened out,  the  octants  being  numbered  from  right  to  left, 
first  above  the  equatorial  or  horizontal  plane  and  then  in  the 
same  order  below,  the  positive  pole  of  the  first  axis,  A,  being 
in  the  middle. 

If,  in  the  form  hkl,  two  of  the  indices  are  equal,  say 
k  =  /,  one  half  of  the  letter  permutations  are  lost,  leaving 

hkk,  kkh,  khk, 

which,  with  the  eight  permutations  of  sign,  gives  a  solid  of 

24  faces,  corresponding  to  two  classes  of  forms,  according  as 

the  two  equal  indices  are  smaller  or  larger  than  the  third. 

In  the  former  case,  when  h>k,  the  resulting  form  is  of  the 

kind  shown  in  fig.  17,  known  as  an 

icositetrahedron,  or,  more  properly, 

trapezoidal  icositetrahedron— from 

the    shape    of    the    faces — which 

is  often  contracted   to   trapezohe- 

dron.     This  may  be   regarded  as 

a  hexakisoctahedron,  in  which  the 

interfacial  angle  over  the  long  edges 

is  1 80°,  or  these  edges  are  effective 

only  as  edges  of  symmetry.     The 

actual    edges     are     therefore     24 

longer  in  the  principal   planes  of 

symmetry,  and  24  shorter  in  the  other  six  planes  of  symmetry. 

The  solid  angles  are,  six  four-faced,  each   formed  by  the 

meeting  of  four  longer  edges  in  the  poles  of  the  principal 

axes  ;  eight  three  faced,  formed  by  groups  of  three  shorter 

edges  in  the  axes  of  ternary  symmetry  ;  and  eight  four-faced 

(two  and  two  edged)  or  formed  by  two  longer  and  two  shorter 

edges  in  the  axes  of  binary  symmetry. 

The  values  of  the  angles  are,  in  the  commonest  kinds, 
2  O  2  and  3  O  3,  or   {21  1}    (3  1 1} 

Over  the  longer  edges .         .   131°  49'     144°  54' 
Over  the  shorter  edges         .   146°  27'     129°  31' 


Systematic  Mineralogy. 


[CHAP.  III. 


Weiss's  symbol  for  this  class  of  form  is  a  :  m  a  :  m  a, 
and  Naumann's  mOm.1     The  construction  of  the  positive 


FIG.  18. 


faces  of  2  O  2  is  shown  in  fig.  18,  the  front  half  of  the 
same  form  in  fig.  17,  and  the  general  arrangement  of  the 
symbols  of  the  whole  of  the  faces  in  fig.  19. 


FIG.  19. 


The  second  case,  when  the  two  equal  indices  are  larger 
than  the  third,  or  as  the  symbol  is  more  conveniently 
written  h  h  k,  is  represented  by  the  class  of  forms  known 

1  The  largest  solid  that  can  be  inscribed  in  a  sphere  symmetrical  to 
the  nine  planes  of  the  cubic  system  is  of  this  class ;  but  it  is  not  a 
possible  crystallographic  one,  m  having  the  irrational  value  2-4142136 
or  tan  67°  30',  and  these  planes  do  not  express  its  full  symmetry. 


CHAP.  III.]  Triakisoctahcdron.  43 

as  triakisoctahedra  (fig.  20).  These  are  contained  by  24 
isosceles  triangles,  whose  shorter  edges  make  eight  three- 
faced  solid  angles  in  the  axes  of 

FIG.  20. 

ternary  symmetry,  and  with  the 
longer  ones  six  eight-faced  (four 
are  four-edged)  solid  angles  in 
the  poles  of  the  principal  axes. 
They  are  particular  forms  of 
hexakisoctahedra,  having  the  di- 
hedral angle  of  the  shorter  edges 
1 80° ;  the  plane  angle  between 
the  two  medium  edges  in  any 
quadrant  of  a  principal  plane  of 
symmetry  is  also  180°,  or  they 

lie  in  the  same  line.  The  effect  of  this  is  to  make  the 
principal  sections  !  square  or  similar  to  those  of  the  regular 
octahedron,  so  that  the  whole  form  may  be  compared  to  an 
octahedron  having  a.  low  three-faced  pyramid  superposed 
upon  each  of  its  faces,  a  property  which  is  indicated  by  the 
name.  The  edges  formed  by  the  meeting  of  these  three 
planes  represent  by  their  position  the  longer  edges  of  the 
hexakisoctahedron. 

The  interfacial  angles  of  the  more  important  of  these 
forms  are — 

Over  the  longer  edges.      Over  the  shorter  edges. 
{221}  2   O  141°  03'  152°  44' 

{331}         3(9  153-28'  142.08'. 

Weiss's  symbol  is  a  :  a  :  m  a,  and  Naumann's  m  O,  signi- 
fying that  one  of  the  parameters  is  m  times  the  unit  or  octa- 
hedral length.  The  construction  for  m  =  2  for  the  faces 
with  positive  indices  is  given  in  fig.  21,  the  front  half  of  the 
same  form  with  the  faces  noted  in  fig.  20,  and  the  general 
scheme  of  notation  for  the  whole  form  in  fig.  22. 

1  The  principal  crystallographic  sections  are  those  upon  planes 
containing  principal  axes  of  symmetry  or  form. 


44 


Systematic  Mineralogy.  [CHAP.  III. 


FIG.  21. 


FIG.  23. 


When  the  three  indices  are  equal,  or  h  =  k  =  i  =  i,  there 
are  no  permutations  of  letters,  and  only  the  eight  sign  per- 
mutations remain,  which  give  the 
regular  octahedron  (fig.  23). 

The  eight  faces  of  this  form 
are  equilateral  triangles,  their  di- 
hedral angle  is  109°  28'  16". 
The  twelve  edges  are  all  equal, 
and  lie  in  the  principal  planes  of 
symmetry,  each  one  representing 
two  of  the  medium  edges  of 
the  hexakisoctahedron.  The  six 
solid  angles,  all  four-faced  and 
similar,  lie  in  the  poles  of  the  axes  of  principal  symmetry. 


CHAP.  III.] 


Octahedron. 


45 


The  axes  of  ternary  symmetry  are  normal  to  the  faces,  and 
those  of  binary  symmetry  normal  to  the  edges.  The  symbols 
are  (i  1 1}.  a  :  a  :  a  and  O. 

The  sections  upon  the  nine  planes  of  symmetry  of  this 
solid,  when  put  together,  as  shown  in  fig.  24,  form  a  skeleton 
octahedron,  made  up  of  forty- 
eight  trihedral  cells,  having  a 
common  apex  in  the  central 
point.  Models  of  this  kind  are 
useful  as  showing  the  relation  of 
the  special  to  the  general  forms  of 
the  system,  and  their  common 
symmetry;  the  same  cellular  ar- 
rangement is  characteristic  of  all, 
the  difference  being  in  the  exter- 
nal contour  of  the  constituent 
planes.  For  the  octahedron  the 
three  principal  sections  are  the 

squares  described  upon  the  edges,  the  other  six  are  rhombs 
having  the  edges  and  crystallographic  axes  for  shorter  and 
longer  diagonals  respectively. 

When  one  of  the  indices  becomes  zero,  or  the  corre-- 
spending  parameter  co-efficient  infinity,  the  change  of  sign 
corresponding  to  that  index  is  lost,  so  that  the  number  of 
sign  permutations  is  only  four,  while  the  full  number  of 
letter  permutations  (six)  are  retained  as  follows  : 

h  ko,  ko  /i,  o  h  k,  o  k  /i,  kho,  h  o  k  ; 

with  the  signs  ++,  H — ,  — h, attached  to  each,  giving 

twenty-four  faces  for  the  complete  form,  which  is  known  as  a 
tetrakishexahedron  {hko}.  Weiss's symbol  is  a  :  ma  :  GO  a, 
and  Naumann's  oo  On.  Fig.  25  shows  the  construction  of 
one-eighth  of  the  form  oo  O  2,  or  {201};  fig.  26  one-half  of 
the  same;  and  fig.  27  the  general  notation  of  the  entire 
number  of  faces.  The  faces  are  isosceles  triangles,  whose 
shorter  sides,  representing  the  longer  edges  of  the  hexakis- 


46 


Systematic  Mineralogy. 


[CHAP.  III. 


octahedron,  meet  in  four-faced  solid  angles  in  the  poles  of 
the  principal  axes,  and  by  threes  with  three  of  the  longer 
sides,  forming  six-faced  solid  angles  in  the  ternary  axes. 
Each  of  these  longer  edges,  representing  two  of  the  shorter 


FIG.  25. 


FIG.  26. 


edges  of  a  hexakisoctahedron,  is  parallel  to  one  of  the  prin- 
cipal axes  of  symmetry,  and  together  they  enclose  a  cube,  so 
that  the  character  of  the  solid  is  that  of  a  cube  with  a  low 


four-sided  pyramid  on  each  of  its  faces.    This  character  is  to 
some  extent  indicated  by  the  name,  and  more  particularly  in 
the  French  cube  pyramid^,  or  the  German  Pyramidenwurfel. 
The  dihedral  angles  of  two  of  the  common  varieties  are — 

Over  the  longer  edges.       Over  the  shorter  edges. 
{210.}       0002  143°  08'  143°  08' 

{310.}     coC>3  126°  52'  154°  09'. 


CHAP.  III.] 


Rhombic  Dodecahedron. 


47 


When  two  of  the  indices  are  equal,  and  the  third  zero, 
which  corresponds  to  a  suppression  either  of  one  change  of 
sign  in  the  triakisoctahedron,  or  of  one  half  of  the  permuta- 
tions in  the  tetfakishexahedron,  the  resulting  form  has  the 
symbols 

h  h  o,  h  o  h,  o  h  h,  with  the  signs  ++,  -\ — ,  — K > 

or  twelve  faces  in  all. 


FIG.  28. 


FIG.  29. 


This  is  the  right  rhombic  dodecahedron,1    {no},  Weiss's 
a  :  a  :  oo  a,  and  Naumann's  GO  O.     Fig.  28  shows  the  con- 


struction from  the  axes  of  the  octahedron  ;  fig.  29  the  front 
half  of  the  form  ;  and  fig.  30  the  symbols  of  all  the  faces. 

1  As  one  index  =  o  and  the  other  two  =  //,  any  whole  number 
may  be  substituted  for  the  latter  without  changing  its  character  ;  or,  in 
other  words,  the  symbol  represents  a  single  form,  and  not  a  series  of 
forms.  The  same  remark  applies  to  the  cube. 


48  Systematic  Mineralogy.  [CHAP.  III. 

Each  of  the  twelve  faces  is  a  rhombus  whose  plane  angles 
are  70°  31'  44"  and  109°  28'  16",  and  its  longer  and 
shorter  diagonals  are  to  each  other  as  \J  2  :  i.  The  dihe- 
dral angles  of  the  faces  are  all  120°,  the  edges  which  repre- 
sent the  longer  ones  of  the  hexakisoctahedron  form  four- 
faced  solid  angles  in  the  poles  of  the  quaternary  axes, 
and  three-faced  ones  in  the  ternary  axes.  Each  parallel 
pair  of  faces  is  normal  to  a  binary  axis,  and  consequently 
the  whole  form  is  parallel  to  the  six  corresponding  planes  of 
symmetry,  the  longer  diagonals  of  the  faces  are  parallel  to 
the  edges  of  an  octahedron,  and  the  shorter  ones  to  those 
of  a  cube.  This  is  one  of  the  solids  having  the  property  of 
filling  up  space — that  is,  it  will  pack  together  with  others  of 
the  same  size  without  leaving  any  hollows. 

Lastly,  when  two  indices  =  o,  the  third  h  may  be  i,  or 
any  other  whole  number,  leaving  only  three  permutations, 
-#00,  o  h  o,  oo  h,  with   +  and  —  signs  to  the  index  //  in 
each,   or   six  in   all.     This   is   the   cube    (fig.   31),   {100} 
FIG  (a  :  oo  a  :  oo  a)    or    <x>  O  oo.       Con- 

sidered as  a  special  case  of  a  hexa- 
kisoctahedron it  is  that  having  only 
the  shorter  edges  effective  as  edges 
of  form,  their  dihedral  angle  being 
90°,  the  longer  edges  are  represented 
by  the  diagonals  of  the  faces  and  the 
medium  ones  by  lines  parallel  to  the 
edges  passing  through  the  centre  of 

the  faces.  The  principal  planes  of  symmetry  correspond  to 
the  axial  planes,  which  are  also  parallel  to  the  faces  ;  this 
form,  therefore,  like  the  rhombic  dodecahedron,  is  sym- 
metrical to  its  own  faces,  or  is  contained  by  its  principal 
planes  of  symmetry. 

The  above  include  all  the  possible  cases  of  holohedral 
cubic  forms,  as  will  be  seen  by  the  following  enumeration, 
in  which  a  different  order  from  that  adopted  in  the  descrip- 
tion is  followed  : 


CHAP.  III.  Naumantfs  Diagram,  49 

1.  hkl.    Hexakisoctahedra.      Indices   all    different,   the 

least  >  o. 

2.  hko.  Tetrakishexahedra.      Indices   all   different,   the 

least  :=  o. 

3.  h  k  k,  Icositetrahedra.     Two  indices  equal,  less  than 

the  third,  and  >  o. 

4.  h  o  o.  Cube.     Two  indices  equal,  less  than  the  third, 

and  =  o. 

5.  hhk.  Triakisoctahedra.  Two  indices  equal,  and  greater 

than  the  third,  which  is  >  o. 

6.  hhQ.  Rhombic   dodecahedron.     Two  indices  equal, 

and  greater  than  the  third,  which  is  =  o. 

7.  h  h  h.  Octahedron.     All  three  indices  equal. 

These  relations  may  also  be  shown  graphically  by  the 
diagram  fig.  32,  using  Naumann's   notation.      The  symbol 
of  the  unit  form,   O,  is  placed  at 
the  summit  of  a  triangle,  the  left- 
hand  side  contains  the  forms  ob-  /  \ 
tained    by    varying    one    of    the                  /       \ 
unit  axes,  or  the  triakisoctahedra,                /  \ 
m  O,  which  pass  downwards  into            m/0^^     ^™A** 
the   rhombic  dodecahedron  oo  O,          /  \ 
when  m  =  GO ;  the  right  side  con-        / 

tains  the  different  icositetrahedra,    «=" °°  07t ~°c° 

m  O  m  similarly  passing  downwards 

into  the  cube  oo  o  GO  ;  when  m  =  oo,  and  the  base  contains 
forms  intermediate  between  the  cube  and  rhombic  dodeca- 
hedron, or  the  tetrakishexahedra  GO  O  n.  The  general 
symbol  of  the  hexakisoctahedra,  mOn,  occupies  the  centre, 
and  the  guide  lines  connecting  it  with  the  sides  indicate  a 
passage  to  the  triakisoctahedra  when  n  =  i,  to  the  icosi- 
tetrahedra when  ;/  =  m,  and  to  the  tetrakishexahedra  when 
m  =  co  . 

He mihedral  forms  of  the  ciibic  system.     These  are  ob- 
tained from  the  holohedral  forms  by  the  symmetrical  sup- 

E 


Systematic  Mineralogy. 


[CHAP.  III. 


pression  of  one-half  of  their  faces,  which  may  be  done  in 
different  ways.  The  relations  of  these  to  each  other  is  best 
shown  by  considering  first  those  derived  from  the  general 
form  {h  k  /}  of  which  all  the  others  are  particular  cases.  In 
this  there  are,  as  already  shown,  six  permutations  of  letters 
and  eight  of  signs,  giving  the  forty-eight  faces.  But  these 
divide  symmetrically  into  four  groups — namely,  the  letter 
permutations  into  the  two  following  cyclical  groups  : 

Direct hkl,  klh,  Ihk-, 

Inverse  .         .         ,         .         .     Ik  /i,  k  h  /,  h  Ik  ; 

and  the  signs  into  two  groups  of  four : 

Direct    .         .     +  +  +,+__,_  +  _,--  +  ; 
Inverse  .         . )_  +  +j+_  +  >+  +  _- 

the  full  arrangement  being  represented  in  the  following 
table  given  by  Miller  : 


A 

B 

hkl 

klh 

lhk~ 

Ikh 

khl 

hlk 

hkl 

klli 

Ihk 

Ikh 

khl 

hlk 

hkl 

klh 

~lhk 

~lkh 

Ih'l 

hlk 

hkl 

~k~lh 

~lhk 

7kh 

khl 

li'lk 

C 

D 

JikT 

k7h 

~lhk 

~lkh 

kliJ 

7i~l~k 

~hkl 

~klh 

Ihk 

Ikh 

khl 

hlk 

hkl 

klh 

Ihk 

Ikh 

khl 

h'lk 

hk'l 

klh 

Ihk 

Ikh 

khl 

hl~k 

There  are  three  ways  in  which  the  four  parts  of  the 
above  table,  distinguished  by  the  letters  A  B  c  D,  can  be 
grouped  so  as  to  divide  the  whole  into  two  symmetrical 
parts,  namely : 

Chequerwise,  or  A  with  D  and  B  with  c  ; 

Into  right  and  left  halves,  or  A  with  c  and  B  with  D  ; 

Into  upper  and  lower  halves,  or  A  with  B  and  c  with  D. 


CHAP.  III.]  PlagiJicdral  Hemihedrism.  51 

The  first  of  these,  corresponding  to  the  two  orders — 

1.  Direct  letters  and  signs  +  inverse  letters  and  signs  (A  +  D), 

2.  Direct  letters  and  inverse  signs  +  inverse  letters  with  direct  signs  (B  +  c) 

— gives  the  forms  represented  in  figs.  33  and  35,  the  former 
being  derived  from  the  extension  of  the  white  faces,  and  the 
latter  of  the  shaded  ones,  in  the  hexakisoctahedron  {321} 
(fig.  34).  It  corresponds  to  an  extension  of  alternate  faces 


FIG.  33. 


FIG.  34. 


FIG.  35. 


of  the  holohedral  form,  so  that  all  the  edges  of  the  latter 
are  obliterated,  the  only  points  common  to  the  original  and 
derived  forms  being  the  ends  of  the  principal  and  ternary 
axes ;  and  as  in  the  form  \h  k  1}  any  face  is  inclined  to  each 
of  the  three  planes  of  symmetry  in  which  its  edges  lie,  at  less 
than  a  right  angle,  it  will  by  extension  lose  its  symmetry  to 
all  of  them,  and  the  resulting  forms  will  be  plagihedral  (skew- 
faced)  or  asymmetric,  which  names  are  used  to  indicate 
this  particular  kind  of  hemihedrism.  There  is,  however,  no 
change  in  the  axes  of  symmetry,  which  are  of  the  same 
number  and  kind  as  in  the  holohedral  forms.  Geometri- 
cally, their  faces  are  irregular  pentagons,  which  are  so 
arranged  that  the  two  correlated  (direct  and  inverse  or 
positive  and  negative  !)  hemihedra,  derived  from  the  same 
holohedral  form,  cannot  be  superposed  or  made  to  corre- 

1  Either  may  be  considered  as  the  positive  or  negative  one  ;  but  the 
choice,  when  made,  applies  to  all  similar  forms  in  the  same  substance. 
Generally  that  containing  the  face  h  k  I  is  taken  as  positive. 

E  2 


52  Systematic  Mineralogy.  [CHAP.  III. 

spond  with  each  other  by  rotation.  It  will  be  seen  that  the 
edges  forming  the  four-faced  solid  angles  at  the  extremities 
of  the  principal  axes  have  a  right-handed  inclination  to  the 
vertical  and  horizontal  lines  in  one  form,  and  a  left-handed 
one  in  the  other,  which  cannot  be  altered  by  change  of 
position.  Such  solids  are  said  to  be  non-superposable,  or 
enantiomorphous,  i.e.  permanently  right-  and  left-handed, 
like  gloves.  This  hemihedrism  is  not  applicable  to  any  of 
the  specialised  forms  of  the  hexakisoctahedron,  or  the  other 
six  holohedral  forms  of  the  system,  for  it  requires  the  four 
groups  of  signs  and  letters  to  be  unbroken,  which  they 
cease  to  be  if  two  letters  are  interchangeable,  or  if  a  sign 
becomes  ambiguous.  This  may  be  seen  in  another  way  by 
considering  each  of  these  six  forms  as  a  hexakisoctahedron, 
in  which  the  dihedral  angle  over  one  or  more  of  the  three 
kinds  of  edges  is  180°,  when,  supposing  one  of  the  faces 
meeting  in  such  an  edge  to  be  removed,  the  extension  of 
the  other  will  only  fill  up  its  place,  and  the  particular  shape 
will  be  restored.  This  would  be  expressed,  in  ordinary 
mineralogical  language,  as  follows  : — The  plagihedral  hemi- 
hedral  forms  of  the  cubic  system  are,  with  the  exception  of 
those  derivable  from  hexakisoctahedra,  undistinguishable 
from  the  holohedral  forms.  These  particular  hemihedral 
forms  are  not  known  to  exist  either  in  natural  or  artificial 
crystals,  and  therefore  no  special  class  of  symbols  are  re- 
quired for  them.  They  may  be  indicated  by  a  \Jikl} 
a  [Ikh],  the  prefix  a  representing  asymmetric. 

Parallel  hemihedrism.  The  second  kind  of  hemi- 
hedrism represented  by  the  division  of  the  table  on  page  50 
into  right-  and  left-hand  halves,  which  corresponds  to  the 
application  of  both  kinds  of  sign  permutations  to  each 
group  of  letter  permutations  taken  separately,  gives  rise  to 
hemihedral  forms  with  parallel  faces.  For  the  hexakis- 
octahedron these  are  shown  in  figs.  36  and  38,  the  former 
being  derived  from  the  white,  and  the  latter  from  the  shaded 
faces  in  fig.  37.  These  are  twenty-four  faced  solids,  known 


CHAP.  III.] 


Parallel  Hcmihedrism. 


53 


as  dyakisdodecahedra ;  the  faces  are  trapezoidal  with  two 
equal  and  two  unequal  sides.  In  each  of  the  principal 
sections  half  the  edges  represent  those  of  the  holohedral 
form  extended,  the  other  half  being  replaced  by  shorter 


FIG.  36. 


FIG.  37. 


FIG.  38. 


ones,  which  are  less  steeply  inclined  to  the  principal  axis 
in  which  they  meet  than  the  former.  The  remaining  edges 
make  three-faced  solid  angles  in  each  of  the  axes  of  ternary 
symmetry,  having  no  symmetrical  relation  to  those  of  the 
holohedral  form.  The  general  result  of  this  is,  that  the 
symmetry  to  the  binary  axes  is  lost  and  that  to  the  principal 
axes  reduced  from  quaternary  to  binary,  while  that  to  the 
ternary  axes  is  unchanged,  and  only  the  three  principal 
sections  remain  planes  of  symmetry.  The  general  symbols 
for  these  forms  are,  for  fig.  36  TT  {Ikh},  and  for  fig.  37 

( 7  7  /I       -NT  )  \~rnO  n~\        j  \~mOti~\     ^ 

IT  \hkl\.     Naumanns  are     • —         and    —  \;  the 

L      2       J  L       2       J 

prefix  TT  and  the  square  brackets  respectively  indicating 
parallel  hemihedrism.  The  positive  and  negative  forms 
derived  from  the  same  hexakisoctahedron  are  superposable, 
or  either  one  may  be  brought  in  the  position  of  the  other 
by  a  quarter  revolution  about  a  principal  axis. 

The  dihedral  angles  in  the  particular  case  given  TT,  (3  2  i } 
are  : 

Over  the  longer  edges  in  the  principal  sections    .  150°  o' 

„        shorter  edges  in  the  principal  sections  .  115°  23' 
„        unsymmetrical   edges   in   the  principal 

sections 141°  47' 


54 


Systematic  Mineralogy. 


[CHAP.  III. 


The  tetrakishexahedron,  when  similarly  developed,  pro- 
duces fig.  39  from  the  white  faces,  and  fig.  41  from  the 
shaded  ones,  in  (2  i  o}  fig.  40.  These  are  pentagonal  dode- 
cahedra  with  irregular  faces,  one  of  the  sides  being  promi- 
nently longer  1  than  the  other  four,  which  are  equal  to  each 


FIG.  39. 


FIG.  40. 


FIG.  41. 


other.  These  longer  sides  are  the  only  effective  edges  in 
the  principal  sections,  and  as  they  are  parallel  to  the 
principal  axes  the  partial  cubic  symmetry  is  at  once  ap- 
parent. The  three-faced  solid  angles  mark  the  symmetry  to 
the  ternary  axes,  and  that  to  the  binary  axes  is  wanting,  as 
in  the  preceding  instance.  The  symbols  are  : 


L  2 


and  - 


_r 


For  the  particular  case  given,  •*  (2  i  o}  ,  the  dihedral  angles 
are  : 

Over  the  longer  edges  .         .        .         .     126°  52' 
Over  the  shorter  edges         .         .         .     113°  35' 

The  dihedral  angle  over  the  edges  in  the  principal 
sections  of  the  hexakisoctahedron  increases  as  the  value  of 
in  in  its  symbol  is  increased,  becoming  180°  when  m  =  oc, 
or  when  it  becomes  a  hexakistetrahedron  ;  the  same  rela- 
tion holds  good  with  its  hemihedral  form,  the  dyakisdode- 
cahedron,  which  under  similar  conditions  passes  into  a 

1  Or  shorter  when  n  approximates  to  i.  The  regular  or  Platonic 
dodecahedron  is  an  impossibly  form,  as  the  indices  of  the  hexakis- 
tetrahedron producing  it  are  V$  +  I,  2,  o,  which  are  not  admissible  on 
account  of  irrationality.  TT  {850}  is  very  near  to  it. 


CHAP.  III.] 


Inclined  Hemihedrism. 


55 


pentagonal  dodecahedron.  In  the  same  way.  when  n  is 
the  symbol  of  the  hexakistetrahedron,  and  its  hemihedral 
form  becomes  GO,  they  pass  into  the  cube. 

This  mode  of  hemihedrism  is  not  applicable  to  the  other 
five  holohedral  forms,  or  rather  it  does  not  produce  geo- 
metrically different  forms  from  them. 

Inclined  hemihedrism.  When  the  faces  are  so  selected 
that  the  whole  of  the  letter  permutations  with  only  one-half 
of  the  sign  permutations  appear  in  a  hemihedral  form,  the 
latter  is  said  to  be  hemihedral  with  inclined  faces.  This 
corresponds  to  the  division  of  the  table  on  page  50  into 
upper  and  lower  halves,  the  former  containing  the  symbols 
with  an  odd  number — one  or  three — of  positive  indices,  and 
the  latter  those  with  one  or  three  negative  indices,  or,  in 
other  words,  a  face  and  its  counterpart  can  never  appear  on 
the  same  form.  Geometrically,  this  signifies  the  extension 
of  all  the  faces  in  alternate  octants,  or  the  suppression  of 
those  in  opposite  and  adjacent  octants  of  the  holohedral 
form,  as  shown  in  fig.  42  for  the  white,  and  in  fig.  44  for  the 


FIG.  42. 


FIG.  43. 


FIG.  44. 


shaded  faces  of  {3  2  i}fig.  43.  This  solid  is  called  a  hexakis- 
tetrahedron, from  its  resemblance  to  a  tetrahedron  enclosed 
by  four  groups  of  six-faced  pyramids.  From  the  construc- 
tion it  will  be  easily  seen  that  the  edges  forming  the  six- 
faced  solid  angles  are  the  same  as  the  longer  and  shorter 
edges  of  the  holohedral  form,  and  preserve  their  character- 
istic inclination,  but  the  new  ones  formed  by  the  meeting  of 
the  extended  faces  have  no  symmetrical  relations  to  the 


Systematic  Mineralogy. 


[CHAP.  III. 


original  form.  The  highest  symmetry  is  therefore  ternary 
about  the  normals  of  the  octahedron,  that  about  the  prin- 
cipal axes  is  reduced  from  quaternary  to  binary,  and  the 
normals  to  the  rhombic  dodecahedron  are  not  axes  of 
symmetry.  As  a  consequence,  the  symmetry  to  the  faces 
of  the  cube  is  lost,  while  that  to  the  faces  of  the  rhombic 
dodecahedron  remains.1 

The  symbols  are  : 

m  On  m  O  n 

K {h k  1}  and  K\hkl\  or  — - —  and  —  — - — • 

The  dihedral  angles  of  the  particular  form  given,  K  (3  2  1} , 
will  be  the  same  as  those  over  the  longer  and  shorter  edges  in 
{321} ;  those  over  the  special  hemihedral  edges  are  120°  55'. 
The  inclined  hemihedral  forms  of  the  icositetrahedron 
are  called  triakistetrahedra,  fig.  45  being  that  derived  from 
the  white,  and  fig.  47  from  the  shaded  faces  of  fig.  46. 


FIG.  45. 


FIG.  46. 


FIG.  47. 


In  this  the  new  edges  formed  by  the  extended  faces  enclose 
a  regular  tetrahedron,  upon  each  of  whose  faces  a  trian- 
gular pyramid  formed  by  the  original  faces  is  superposed. 
The  symbols  are  : 

K  \h  k  k}  and  K  {hkk},  or  and . 

For  the  particular  case,  K  (2  1 1} ,  the  dihedral  angles  over  the 

1  As  these  relations,  which  hold  good  for  all  inclined  hemihedral 
forms,  are  not  readily  seen  in  perspective  figures,  it  will  be  well  for  the 
student  "to  study  them  upon  a  model.  The  regular  tetrahedron  is  the 
most  convenient  form  for  this  purpose. 


CHAP.  III.] 


Inclined  Hem  ikedrism . 


57 


tetrahedral  edges  are  109°  28',  or  the  supplements  of  those 
of  the  regular  tetrahedron;  those  over  the  shorter  edges  have 
the  same  value  as  in  {211}. 

The  inclined  hemihedral  forms  of  the  triakisoctahedron 
are  called  deltoid  dodecahedra  (figs.  48  and  50),  of  which 
the  first  is  derived  from  the  white,  and  the  second  from  the 
shaded  faces  in  (2  2  1}  fig.  49.  In  these  the  extended  faces 


FIG.  49. 


FIG.  50. 


form  a  new  three-edged  pyramid  above  the  enclosed  octa- 
hedron, but  of  different  proportions  to  those  of  the  original 
form,  the  faces,  when  regularly  developed,  being  deltoids  or 
four-sided  figures,  one  of  whose  diagonals  joins  similar  and 
the  other  dissimilar  angles.  The  symbols  are  : 

mO  mO 

K{h/ik\  and  K  {hhk},  or and  — . 

The  dihedral  angles  over  the  hemihedral  edges  in  K  {2  2  1} 
are  right  angles. 

The  octahedron  (fig.  52)  by  this  method  produces  the 


FIG.  51. 


FIG.  52. 


FIG.  53. 


two  regular  tetrahedra  (figs.  51  and  53)  as  its  hemihedral 
forms.      Their  faces   are  all   equilateral   triangles  of  four 


Systematic  Mineralogy. 


[CHAP.  III. 


times  the  area  and  twice  the  length  of  edge  of  those  of  the 
octahedron  from  which  they  are  derived.  The  dihedral 
angles  are  70°  32',  or  supplements  to  those  of  the  octa- 
hedron ;  the  edges  correspond  to  diagonals  of  the  faces  of 
a  cube  described  about  the  axes  of  the  enclosed  octahedron. 
The  principal  axes  join  the  middle  points  of  opposite  edges ; 
the  ternary  axes  are,  as  in  the  holohedral  form,  normals  to 
faces,  but  on  one  side  only,  the  opposite  sides  meeting  the 
extended  faces  in  the  solid  angles. 

Neither  the  cube  nor  the  rhombic  dodecahedron  can 
produce  forms  dissimilar  from  themselves  by  any  of  the 
three  methods  of  hemihedrism.  The  former,  when  con- 
sidered as  a  particular  form  of  hexakisoctahedron,  has 


FIG. 


FIG.  56.  , 


eight  of  its  faces  in  the  same  plane,  and  therefore  the 
removal  of  half  of  these,  whether  by  alternating  eighths, 
as  in  fig.  54,  or  by  quarters,  as  in  figs.  55,  56,  which  corre- 
spond to  the  three  methods  of  hemihedral  selection,  can 


FIG.  57. 


FIG.  58. 


FIG.  59. 


have  no  effect,  as  the  extension  of  the  remaining  parts, 
whether  white  or  shaded,  will  restore  the  original  form.  In 
the  same  way  the  rhombic  dodecahedron  in  each  of  its 


CHAP.  III.]  Hem  iJiedral  D  iagrams.  5  9 

faces  contains  four  of  those  of  [h  kl}  ;  and  therefore  their 
removal  by  quarters  or  halves,  as  in  figs.  57,  58,  and  59,  and 
the  extension  of  the  remainder,  does  not  change  the  form. 

It  may  therefore  be  said  that  there  are  no  hemihedral 
forms  of  these  solids  ;  but  this  is  only  true  as  a  geometrical 
proposition,  and  is  at  variance  with  a  general  principle, 
deduced  from  observation,  that  the  forms  making  up  the 
crystals  of  any  particular  substance  are  all  of  the  same  kind 
— i.e.  either  all  holohedral  or  of  the  same  class  of  hemi- 
hedrism.  For  example,  in  iron  pyrites  a  very  large  number 
of  forms  are  known,  the  principal  ones  being  pentagonal-  and 
dyakis-dodecahedra,  which  often  appear  alone  as  well  as  in 
combination  with  the  cube  and  octahedron,  but  never  with 
a  hexakisoctahedron  or  tetrakishexahedron,  or  any  inclined 
hemihedrai  form.  It  is  therefore  necessary  to  consider  the 
cube  and  rhombic  dodecahedron  as  common  to  the  holo- 
hedral and  hemihedral  classes  of  forms  alike,  their  true 
character  being  only  determinable  by  the  nature  of  their 
combinations.  The  relation  and  derivation  of  the  different 
hemihedral  forms  may  be  expressed  by  their  symbols,  ac- 

FIG.  60.  FIG.  61. 

to 

±  a 

/\ 


cording  to  Naumann,  in  diagrams  similar  to  that  already 
given  for  the  holohedral  form.  Fig.  60  gives  the  scheme  for 
the  parallel,  and  fig.  61  for  the  inclined,  class. 

In  Weiss's  notation,  parallel  hemihedral  forms  are  dis- 
tinguished by  the  symbol  \  preceding  that  of  the  type 
face;  and  the  inclined  ones  in  the  same  way,  with  the 


6o 


Systematic  Mineralogy.  [CHAP.  III. 


addition  of  a  positive  or  negative  sign.  Thus  the  two 
dyakisdodecahedra,  TT  \Jikl}  and  TT  {Ikh},  are  respectively 
\  (a  :  ma  :  no)  and  i  (na  :  ma  :  a),  and  the  hexakis- 
tetrahedra,  K  {hkl}  and  K  {hk  /}  +  \(a  \  n  a  \  ma)  and 
—  ^  (a  :  n  a  :  m  a). 

Tetartohedral  forms.  If  any  hemihedral  form  of  the 
hexakisoctahedron  be  subjected  to  one  of  the  other  modes 
of  hemihedry,  a  new  form  containing  only  one-fourth  of  the 
full  number  of  faces,  but  otherwise  satisfying  the  general 
conditions  of  systematic  symmetry,  is  obtained.  This  is 
said  to  be  tetartohedral.  By  reference  to  the  table  on 
page  50  it  will  be  easily  seen  that  the  result  is  the  same 
whichever  methods  of  hemihedry  be  employed,  the  whole 
series  of  symbols  being  resolved  into  the  four  groups,  each 
containing  one  group  of  letter  and  sign  permutations,  and 
representing  a  separate  form.  Of  these,  only  two  are  geo- 
metrically distinct  otherwise  than  by  position  ;  that  contain- 
ing direct  letters  with  inverse  signs  is  superposable  to  that 
containing  inverse  letters  with  direct  signs,  or  B  with  c,  as 
are  also  the  groups  A  and  D,  where  letters  and  signs  are 
both  direct  or  both  inverse,  but  not  otherwise — that  is,  the 
tetartohedra  resulting  from  the  same  hemihedron  are  right- 
and  left-handed  to  each  other. 

Figs.  62  and  63  represent  the  forms  obtained  by  the 


FIG.  62. 


FIG.  63. 


application   of  parallel  hemihedrism   to   the   hexakistetra- 
hedron  (fig.  42).     The  first  corresponds  to  the  white  faces 


CHAP.  III.]  Cubic  TetartoJiedrism.  6l 

and  the  second  to  the  shaded  ones,  in  alternate  (the  ist,  3rd, 
6th,  and  8th)  octants  of  the  hexakisoctahedron  (fig.  34),  or 
to  the  divisions  A  and  B  of  the  table  on  page  50.  These 
are  called  right  and  left  tetartohedral  pentagonal  dodeca- 
hedra,  their  faces,  when  most  regularly  developed,  being 
irregular  pentagons.  The  only  points  in  common  with  the 
holohedral  forms  are  alternate  extremities  of  the  ternary 
axes,  which  retain  their  original  positions  :  those  of  the 
principal  axes  meet  in  six  edges  which  are  oblique  to  the 
principal  sections.  As  both  kinds  of  hemihedrism  are 
involved  in  their  production,  the  symmetry  to  both  series  of 
planes  is  lost,  or  the  forms  are  plagihedral,  but  that  to  the 
axes  is  of  the  same  kind  as  in  the  hemihedral  forms,  or 
to  three  binary  and  four  ternary  axes.  This,  therefore, 
may  be  regarded  as  the  fundamental  axial  symmetry  of  the 
cubic  system,  it  being  the  minimum  common  to  all  classes 
of  forms,  the  higher  quaternary  kind  being  added  in  the 
case  of  the  holohedral  and  plagihedral  hemihedral  classes. 

The  inverse  tetrakishexahedron  gives  rise  to  two  similar 
forms  differing  only  in  position,  and  if  the  former  be  con- 
sidered as  positive  they  will  be  the  negative  tetartohedra. 
The  general  symbols  will  be  : 


/  }        7T  K  (7  k  /l] 

and 

m  On  m  O  n 

* 


. 

T 


4  4 

m  On  m  O  n 

•• 


These  are  the  only  geometrically  distinct  tetartohedral 
forms,  and,  although  crystallographically  possible,  they  are 
not  known  to  occur  independently  in  either  natural  or  arti- 
ficial crystals.  The  actual  existence  of  the  condition  of 
tetartohedrism  is,  however,  known  from  the  fact  of  a  few 
substances  appearing  in  crystals  showing  both  kinds  of 


62 


Systematic  Mineralogy.  [CHAP.  III. 


hemihedrism— a  point  that  will  be  considered  in  treating  of 
the  combinations  of  the  system. 

Combinations  of  the  cubic  system.  Any  number  of  simple 
forms  may  appear  with  their  full  number  of  faces  in  a  single 
crystal,  which  will  then  have  a  sort  of  composite  character, 
the  faces  of  the  constituent  forms  being,  as  a  rule,  recog- 
nisable not  by  their  shapes,  but  by  their  positions.  Such 
crystals  are  termed  combinations.  That  the  shapes  of  the 
faces  must  be  altered,  even  with  the  most  exact  regularity  of 
position,  will  be  apparent  when  it  is  considered  that  the 
simple  forms  described  about  any  definite  lengths  of  axes 
are  all  exterior  to  the  octahedron,  or  enclosed  by  the  cube, 
having  only  points  or  lines  in  common,  and  therefore  no 
combination  is  possible  between  them  in  this  condition. 
If,  however,  the  size  of  any  one  of  them  be  altered  rela- 
tively to  another  by  shifting  its  faces  parallel  to  the  original 
positions,  either  nearer  to  the  centre  in  the  case  of  an 
exterior,  or  further  from  it  in  an  interior  form,  which  does 
not  alter  its  crystallographic  significance,  a  new  solid  will 
be  produced,  with  edges  and  solid  angles  either  wholly  or 


FIG.  64. 


FIG.  65. 


partly  different  from  those  of  the  component  forms.  For 
instance,  fig.  64  represents  the  complete  interpenetration  of 
a  cube  by  an  octahedron,  their  principal  axes  being  common 
as  regards  position,  but  differing  in  length ;  the  points  of 
the  latter  form  project  as  four-faced  pyramids  above  the 


CHAP.  III.] 


Cube  and  Octahedron. 


faces  of  the  former,  and  similarly  the  solid  angles^  of  the 
cube  form  three-faced  pyramids  above  the  faces'  of  the 
octahedron.  If  these  projecting  portions  be  removed,  which 
is  necessary  to  produce  convex  angles,  the  form  is  reduced 
to  fig.  65,  which  may  be  considered  as  an  octahedron  with 
its  solid  angles  cut  off  or  truncated  l  by  the  faces  of  a  cube. 
Fig.  66  is  another  example  in  which  the  faces  of  the  two 
forms  are  evenly  balanced,  so  that  it  may  be  equally  well 


FIG.  66. 


FIG.  67. 


regarded  as  a  cube  modified  by  an  octahedron,  or  the 
reverse ;  while  in  fig.  67  the  cube  is  the  principal  or  domi- 
nant form,  the  octahedron  being  only  represented  by  a 
small  triangular  plane  in  each  of  the  corners.  In  neither 
of  these  are  the  characteristic  shapes  of  the  faces  seen  in 
the  simple  forms  apparent,  but  they  are  nevertheless  easily 
recognisable  from  their  constant  position,  and  the  paral- 
lelism of  the  new  edges  of  combination. 

As  a  rule,  special  names  are  not  applied  to  combinations, 
but  they  are  described  by  their  symbols,  that  of  the  most 
prominent  form  being  placed  first.  Thus,  fig.  65  is  indi- 
cated by  the  symbols  O .  <x>  O<x>,  and  fig.  67  as  oo  O<x> .  O, 
while  either  order  applies  equally  well  to  fig.  66. 

The  faces  of  the  octahedron  when  in  combination  with 
the  rhombic  dodecahedron  appear  as  triangular  planes 

1  An  edge  replaced  by  a  single  plane  making  equal  angles  with  the 
adjacent  faces  is  truncated  ;  when  the  replacement  is  by  two  planes  it 
is  beveled.  Similarly,  a  solid  angle  is  truncated  by  a  single  plane, 
bevelled  by  two,  and  acuminated  or  blunted  by  three  or  more. 


64 


Systematic  Mineralogy. 


[CHAP.   III. 


truncating  the  three-faced  solid  angles  of  the  latter,  as  in 
fig.  68.  When  the  octahedron  is  the  dominant  form,  the 
faces  of  the  rhombic  dodecahedron  truncate  its  edges  as  in 


FIG. 


FIG.  69. 


fig.  69.  The  cube  truncates  the  four- faced  solid  angles  of 
the  rhombic  dodecahedron,  as  in  fig.  70,  and  conversely  the 
latter  truncates  the  edges  of  the  former,  as  in  fig.  71. 


FIG.  70. 


FIG.  71. 


Examples  of  these  combinations  are  of  very  common  occur- 
rence in  nature,  especially  in  magnetite,  galena,  and  fluor- 
spar. 

The  faces  of  the  icositetrahedron,  2  O  2  or  (2  i  i},  lie  in 
zones  whose  axes  are  parallel  to  the  edges  of  the  octahedron 
and  the  plane  diagonals  of  the  cube,  and  therefore  form 
blunt  four-faced  pyramids  upon  the  solid  angles  of  the 
former  (fig.  72),  and  three-faced  ones  upon  the  latter  (fig.  73). 


CHAP.  III.]      Combinations  of  IcositetraJiedron. 


In  a  combination  of  these  three  forms,  when  the  first  pre- 
dominates, as  in  fig.  74,  the  cube  truncates  its  four-faced, 
and  the  octahedron  its  three-faced,  solid  angles. 

The  combinations  of  icositetrahedra  with  the  rhombic 
dodecahedron   vary  very  considerably  in  appearance,    ac- 


FIG.  72. 


FIG.  73. 


CJJD 


AJ 


cording  to  the  value  of  m  or  the  relation  of  h  to  k  in  their 
symbols.  The  form  most  commonly  observed,  which  is 
202  or  {211},  truncates  the  whole  of  the  edges  of  {no};  as 
the  two  forms  are  tautozonal  (fig.  75).  This  combination 


FIG.  74. 


FIG.  75 


gives  a  good  example  of  the  determination  of  the  symbols 
of  a  form  from  those  of  its  zones.  If  the  edges  of  the  cube 
and  octahedron  be  added,  as  shown  by  the  dotted  lines,  it  will 
be  apparent  that  any  face  such  as  the  upper  one,  marked  0, 

F 


66 


Systematic  Mineralogy.  [CHAP.  III. 


lies  in  two  zones,  the  first  containing  i  o  i  and  o  i  i  or  [i  i  i] 
and  the  second  o  o  i  and  i  i  i  or  [i  i  o].  From  these  sym- 
bols we  get,  by  the  method  given  on  page  32,  a  symbol 
(7  T  2)  or  the  counterpart  of  that  of  the  particular  face,  and 
similarly  (:>  1 1)  for  the  second  face,  marked  a.  In  such  a 
case,  therefore,  no  special  calculation  is  required  for  the 
determination  of  the  form,  when  the  parallelism  of  the 
edges  is  exhibited,  by  measurement. 

When  m  >  2  or  //  ;  k  >  2  :  i,  the  faces  of  the  icositetra- 
hedron  appear  as  deltoids  in  groups  of  four  upon  the  four- 


FIG.  76, 


FIG.  77 


FIG.  78' 


faced  solid  angles  of  the  rhombic  dodecahedron,  as  shown 
for  the  combination  oc  O,  3  O  3,  or  {i  i  o} ,  {31 1}  in  fig.  76. 
When  m  <  2  or  h  :  k  <  2  :  i,  similar  planes  appear  in 
groups  of  three  upon  the  three-faced 
solid  angles,  as  in  the  combination 
ootf.f  <?f  or  {no},  {3  2  2}  (fig.  77). 
The  triakisoctahedron,  having  its 
longer  edges  in  common  with  the 
octahedron^  bevels  the  edges  of  the 
'latter  and  conversely    the    octahe- 
dron truncates  the  three-faced  solid 
angles  of  the  former,  both  of  which 
relations  are  apparent  ifi  fig.  78,  con- 
taining the  forms  0 .2  O  or  {i  i  i], 
{221}.     In  combination  with  the  rhombic  dodecahedron  a 


CHAP.  III.]      Combinations  of  Triakisoctahedron.         67 


triakisoctahedron  forms  obtuse  pyramids  blunting  the  three- 
faced  solid  angles  (fig.  79),  and  with  the  cube  groups  of 
three  deltoid  planes  upon  the  solid  angles  (fig.  80). 

FIG.  79. 


FIG,  81. 


The  triakisoctahedron,  in  combination  with  the  icosi- 
tetrahedron,  modifies  the  edges  lying  in  the  dodecahedral 
planes  of  symmetry  (the  short  edges  of  h  k  /).  In  the  par- 
ticular case  represented  in  fig.  81, 
202,  $O,  or  (211),  {332},  the 
two  forms  are  tautozonal,  and 
therefore  the  new  edges  are  parallel 
to  the  original  ones.  When  ;;/  is 
greater  than  f ,  or  the  form  is  nearer 
to  a  rhombic  dodecahedron,  the  re- 
placing faces  appear  as  very  acute 
triangles  whose  summits  meet  in 
the  ternary  axes. 

The  tetrakishexahedron  having 
its  longer  edges  in  <-ommon  with  the  cube,  will  in  com- 
bination bevel  these  edges,  and  conversely  the  cube  will 
truncate  the  four-faced  solid  angles  of  the  first  form,  as 
shown  in  fig.  82,  representing  the  combination  oo  O  oo  . 
QO  O 2  {160},  or  {210},  which  is  commonly  observed  in 
fluorspar.^  The  same  forms  being  also  tautozonal  to  the  rhom- 
bic dodecahedron,  the  latter  will  have  its  four-faced  solid 


Systematic  Mineralogy.  [CHAP.  III. 


angles  blunted  by  the  faces  of  oo  O  2  (fig.  83).  If  the  cube 
were  to  be  added  its  faces  would  truncate  these  new  solid 
angles,  and  any  more  obtuse  tetrakishexahedron  would  bevel 
the  edges  between  {2  i  o}  and  (i  o  o},  and  any  more  acute 
one  those  between  (2  i  o}  and  {i  i  o}. 


F:G.  82. 


\          =«>i        X 

:fo              ion 

210 

t&          V 

FIG.  84. 


The  tetrakishexahedron  and  octahedron  are  not  tauto- 
zonal  forms,  and  therefore  in  combination  their  faces  assume 
irregular  shapes,  as  seen  in  fig.  84.     By  comparing  this  with 
figs.  65  and  69  it  will  be  seen  that 
these  shapes  will  vary  with  the  value 
of  in  or  the  inequality  of  //    and 
k  in  the  symbols  Of  the  tetrakishexa- 
hedron.    As  these  values  diminish, 
the  octahedral  faces  become  more 
nearly  triangular,  until  when  m  =  i 
or   //  =  k  they   are  equilateral   tri- 
angles, as  in  fig.   69 ;  and   in    the 
reverse  direction,  when  m  =  oo   or 
//  :  k  =  i  :  o,  they   are  equiangular 

hexagons  of  120°,  as  in  fig.  65.  The  first  of  the  cases,  how- 
ever, is  the  condition  of  a  rhombic  dodecahedron,  and  the 
second  that  of  a  cube,  which  are  the  respective  limiting  forms 
of  the  tetrakishexahedron  in  these  directions.  The  acute 
angles  of  the  deltoid  planes  in  fig.  84  will  alter  in  a  cor- 
responding manner  to  a  minimum  of  o°  in  fig.  69,  and  to  a 


CHAP.  III.]     Combinations  of  Hexakisoctahedron.         69 


maximum  of  90°  in  fig.  65.  By  comparing  analogous 
combinations  between  limiting  forms,  their  relations  may 
often  be  more  readily  appreciated  than  by  the  most  elabo- 
rate verbal  explanations ;  and  therefore  the  construction 
of  forms  with  different  parameters  to  those  given  in  the 
figures  may  be  recommended  to  the  learner  as  a  useful 
exercise. 

The  combinations  of  the  tetrakishexahedron  with  the 
other  twenty-four-faced  forms  vary  very  considerably  in 
appearance,  accordingly  as  different  values  are  assigned  to 
the  variable  parameters  in  their  symbols,  and  to  illustrate 
them  properly  a  larger  number  of  figures  would  be  required 
than  can  be  given  here.  One  special  case  deserves  men- 
tion— namely,  that  by  regular  truncation  of  the  edges  lying 
in  the  principal  sections,  the  icositetrahedron  (2  i  1}  may  be 
changed  into  the  tetrakishexahedron  (2  i  o} . 

The  hexakisoctahedron  appears  in  combination  with  the 
octahedron  in  groups  of  eight  triangular  faces,  blunting  the 


FIG.  85. 


FIG. 


solid  angles  (fig.  85),  and  in  similar  groups  of  six  faces  upon 
the  solid  angles  of  the  cube  (fig.  86).  From  these  it  will 
also  be  apparent  that  the  octahedron  truncates  the  six-faced, 
and  the  cube  the  eight-faced,  solid  angles  of  the  hexakisocta- 
hedron. 

In  combination  with  the  rhombic  dodecahedron,  the 
hexakisoctahedron  may  appear  either  as  bevelling  the  edges 
or  as  modifying  the  three-  or  four-faced  solid  angles.  The 


Systematic  Mineralogy.  [CHAP.  III. 


first  case  is  that  in  which  the  forms  are  tautozonal,  repre- 
sented in  fig.  87,  for  {i  i  o},  (3  2  1}  or  GO  0, 3  O  |  corresponds 
to  the  condition  h  =  k  +  I  in  {h  k  1}  or  m  n  =  m  +  n,  or 

in  m  0  n.     The   known  forms  of 


m  m 

n  = n.   =    

m  —  i  m  +  i 


this  character  are  {3  2  1} .  {4  3  1}  and  {6  4.63.1},  the  first 
being  that  most  frequently  observed  in  nature.     When 

m 


h  <  k  +  /,  n  < 


or  m  n  <  m  + 


as  in  {543}  .  {432}  or  {15.11.7},  the  replacement  is  by 
six  faces  upon  the  three-faced  solid  angles  of  {i  10}  ;  and 
lastly,  when 


h  >  k  +  I,  n  > 


m  —  i 


or  m  n  >  m  +  n, 


as  in  (42  1} .  {73  i},  &c.,  the  replacement  is  by  eight  faces 
upon  the  four-faced  solid  angles  of  {i  i  o}.  By  far  the  larger 


FIG.  87. 


FIG. 


number  of  known  hexakisoctahedra  are  of  the  latter  kind, 
but  they  are  not  easily  recognised,  as  they  only  occur  in 
very  subordinate  combinations  in  crystals  containing  nume- 
rous other  forms.  For  such  complex  combinations  the  reader 
is  referred  to  Schraufs  Atlas,  and  the  larger  treatises  on 
Mineralogy. 

Hemihedral  combinations.  Among  the  simpler  and  more 
important  cases,  that  of  the  cube  and  pentagonal  dode- 
cahedron is  seen  in  fig.  88.  This  differs  from  fig.  82, 


CHAP.  III.] 


Hem  ihedral  Com  binations. 


by  the  omission  of  the  alternate  bevelling  planes  in  each 
zone,  or  the  faces  of  TT  (2  i  o}  truncate  the  edges  of  {100} 
unsymmetrically,  being  unequally  inclined  to  adjacent  faces  ; 
but  this  inequality  diminishes  with  that  between  h  and  k 
or  k  and  /,  or  as  we  approach  the  cube  and  rhombic  dode- 
cahedron respectively.  This  class  of  combination  is  very 
characteristic  of  the  pyrites  group  of  minerals.  In  fig.  89, 
the  pentagonal  dodecahedron  inverse  to  that  in  fig.  88 
modifies  the  solid  angles  of  the  octahedron  symmetrically. 
If  these  planes  be  extended  to  the  complete  obliteration  of 
the  octahedral  edges,  a  solid  is  obtained  with  twenty  faces 
which  are  very  nearly  equilateral  triangles,  and  approxi- 
mating in  appearance  to  the  regular  icosahedron,  which, 
however,  we  have  seen,  is  not  possible  in  crystallography.1 

The  dyakisdodecahedron   appears  as  an  obtuse  three- 
faced  pyramid  upon  the  ternary  solid  angles  of  the  pentagonal 


FIG.  89. 


FIG.  90. 


\ 


dodecahedron,  and  has  its  own  solid  angles  of  the  same  kind 
truncated  by  the  octahedron,  as  in  fig.  90.  It  also  forms 
groups  of  three  irregular  four-sided  planes  upon  the  solid 
angles  of  the  cube,  as  in  fig.  91. 

In  the  inclined  hemihedral  forms,  the  cube  truncates  the 
edges  of  a  tetrahedron  (fig.  91),  and  has  half  its  own  solid 
angles,  or  one  in  each  pair  joined  by  a  ternary  axis  truncated 
by  the  faces  of  the  latter  (fig.  93).  Two  opposite  tetrahedra 
in  combination  appear  as  in  fig.  94,  the  faces  of  one  being 

1  1 1  i  \  v  [8  5  o\  or  0.  \  [oo  0  f]  is  a  very  close  approximation, 
possible  but  not  actually  observed. 


72  Systematic  Mineralogy.  [CHAP.  III. 

prominently  larger  than  the  other.     It  will  be  easily  seen 
that  when  they  are  equally  developed  the  combination  will 


FIG.  91. 


FIG.  92. 


FIG.  93. 


be  an  octahedron,  and  therefore  indistinguishable  from  a 

holohedral   form.     When    this    does 

occur,   the    hemihedral    nature  of  a 

substance  is  often  apparent  from  some 

physical  dissimilarity  in  the  two  classes 

of  faces,  such  as  one  set  being  more 

brilliant   than   the   other,  or   striated 

when    the   others   are    smooth,    and 

therefore  indicating  that  the  crystal  is 

not  a  true  octahedron. 

The  rhombic  dodecahedron  forms  a  low  three-faced 
pyramid  upon  each  of  the  solid  angles  of  a  tetrahedron,  as 
in  fig.  95. 

FIG.  94.  FIG.  95. 


The  triakistetrahedron  bevels  the  edges  of  the  tetra- 
hedron of  the  same  direction  as  in  the  combination  x  (21  t} 
K  (i  i  1}  (fig.  96),  which  is  very  characteristic  of  the  antimo- 
nial  copper  ore  known  as  fahlerz. 


CHAP.  IV.]          Tetartohedral  Combinations. 


73 


Fig.  97  represents  one  of  the  few  known  cases  of  cubic 
tetartohedrism,  as  evidenced  by  the  occurrence  of  inclined 


FIG.  96. 


FIG.  97. 


and  parallel  hemihedral  forms  in  the  same  combination.  It 
is  not  a  natural  substance,  being  a  crystal  of  an  artificial 
salt — chlorate  of  sodium.  The  same  class  of  development 
characterises  the  crystals  of  the  nitrates  of  lead,  strontium,  and 
barium,  an  actual  tetartohedron  +  \  (5  O  f )  r,  or  ^{351}, 
having  been  determined  by  Lewis  in  the  latter  salt. 


FIG.  98. 


CHAPTER   IV. 

HEXAGONAL   SYSTEM. 

THE  symmetry  characteristic  of  the  most  completely  de- 
veloped forms  in  this  system  is  seen  in  the  regular  hexagonal 
prism  with  parallel  end  faces 
(fig.  98).  This  has  binary  sym- 
metry about  six  axes,  making 
angles  of  30°  and  150°  to  each 
other,  the  strong  lines  #,,  <z2,  a3, 
and  the  dotted  ones  alternat- 
ing with  the  same  plane,  and 
senary  or  hexagonal  about  a 
seventh  axis,  which  is  normal  to 
the  other  six,  and  indicated  by 
the  vertical  line  c.  These  axes 
correspond  to  seven  planes  of 
symmetry  :  a  principal  one,  or  that  containing  the  six  binary 


74  Systematic  Mineralogy.  [CHAP.  IV. 

or  lateral  axes,  normal  to  the  principal  or  vertical  axis,  and 
six  others  corresponding  to  the  diametral  sections  of  the 
prism  upon  each  of  the  lateral  and  the  vertical  axes.  Three 
of  these,  or  those  parallel  to  the  faces  of  the  prism,  are  dis- 
tinguished as  lateral  axial  planes,  and  the  alternate  ones 
containing  the  dotted  lateral  axes  as  lateral  interaxial  planes. 
The  geometrical  relations  of  faces  of  the  above  kind  may 
be  expressed  in  four  different  ways,  each  of  which  has  been 
adopted  as  the  basis  of  a  system  of  notation.  These  are  : 

1.  Weiss 's  system.^-    With  four  reference  axes,  three  lateral 
at  60°  and  120°  to  each  other,  and  perpendicular  to  a  fourth 
or  principal  axis,  taken  in  the  order  of  fig.  98,  or  alf  az,az,  c. 

2.  Schraufs  system.     If  in  fig.  98  the  faces  meeting  the 
right  and  left  lateral  axis  a%  be  produced  until  they  meet 
both  in  front  and  behind,  the  result  is  a  four-faced  prism 
upon  a  rhombic  base,  whose  diagonals  are  at  right  angles, 
and  in  the  ratio  of  i  :  >/3,  or  that  of  the  side  of  an  equi- 
lateral triangle  to  its  altitude.    This  therefore  gives  a  method 
whereby  hexagonal  forms  may  be  referred  to  three  indepen- 
dent axes  all  at  right  angles  to  each  other,  as  in  the  rhombic 
system;  if  the  constant  relation  of  i  :  «/3  be  assumed  for 
the  lateral  axes.     This  method  is  adopted  by  Schrauf,  who 
calls  it  the  orthohexagonal  system  ;  by  it  the  prism  in  fig.  98 
is  not  a  simple  form,  but  a  combination  of  the  unit  rhombic 
prism  {i  i  o},  and  a  form  containing  two  faces  parallel  to  the 
right  and  left  diametral  plane,  or  {i  oo}. 

3.  Millers  system.     If  a  cube  be  placed  with  a  ternary 
axis  upright,  the  three  edges  meeting  in  either  of  the  poles 
of  that  axis  will  be  parallel  and  opposite  to  the  three  at  the 
other  pole,  so  that  if  lines  parallel  to  them  be  drawn  through 
the  centre  a  system  of  three  axes  will  be  obtained,  inclined  to 
the  vertical,  but  making  equal  angles  with  each  other,  and 
having  equal  parameters.    In  this  particular  case  these  angles 
will  be  right  angles,  but  the  same  relation  holds  good  for 
any  analogous  form  contained  by  six  equal  rhombic  faces, 

1  Weiss  calls  the  system  sechsgliedrig,  or  six-membered. 


CHAP.  IV.]  Hexagonal  Notation.  75 

rhombohedron,  whose  solid  angles  are  formed  by  edges  meet- 
ing at  a  greater  or  less  angle  than  90°,  which  inclination 
will  also  be  characteristic  of  their  axes.  The  relation  of 
such  a  system  of  axes  to  the  unit  faces  is  that  of  the  legs 
of  a  table  formed  by  three  sticks,  tied  together  in  the 
middle,  to  the  table  top  and  the  ground  respectively— the 
first  being  the  face  i  i  i,  and  the  second  i  i  i.  This  prin- 
ciple is  adopted  in  Miller's  rhombohedral  notation. 

4.  Bravais- Miller  system.  The  three  semi-axes,  having 
like  signs  in  Miller's  system,  when  represented  by  their  ortho- 
gonal projections,  are  resolved  into  a  common  vertical  and 
three  horizontal  lines,  the  latter  making  angles  of  120°  with 
each  other.  This  brings  us  back  to  the  system  of  four  axes 
with  the  difference  that  the  positive  and  negative  semi-axes 
alternate  with,  instead  of  succeeding,  each  other  at  60°  in 
the  horizontal  plane  ;  and  in  this  way  we  obtain  the  hexa- 
gonal notation  of  Bravais  as  adapted  to  Miller's  system. 
It  has  the  advantage  of  maintaining  the  relation  between 
the  notation  by  indices  and  that  by  parameter-coefficients 
subsisting  in  the  other  systems,  as  well  as  of  expressing  the 
physical  symmetry  of  the  forms  more  readily  than  the  other- 
wise preferable  rhombohedral  notation  of  Miller.  The  pro- 
perties of  any  face  are  determined  by  three  indices,  one 
referring  to  the  independent  vertical  axis  and  the  others 
to  two  of  the  lateral  axes  ;  but 
to  determine  its  position  in  the 
form,  a  fourth  index,  referring  to 
the  third  lateral  axis,  is  required, 
giving  a  general  symbol  of  the 
form  {/i  kit],  in  which  the  posi- 
tion of  the  last  letter,  referring 
to  the  vertical  axis,  is  invariable,  while  the  other  three  are 
interchangeable  with  positive  and  negative  signs,  subject  to 
the  condition  that  their  algebraical  sum  is  always  equal  to 
zero,  or  h  +  k  +  /  =  o.  This  property  will  be  apparent 
from  fig.  99,  where  O  x,  Oy,  and  O  z  represent  the  distances 


76  Systematic  Mineralogy.  [CHAP.  IV. 

at  which  a  right  line  cuts  three  axes,  making  angles  of  60° 
to  each  other  conjointly  at  O. 

For, 

&Oxz  +  AO  zy  +  A  Oyx  =  o, 
or, 

Ox.  O  z  sin.  xO  z  +  Oz.  Oy .  sin.  z  Oy  +  Oy  Ox 

sin.  yOx  =  o; 
but  since 

the  sines  are  all  equal.  Dividing  them  out,  and  also  dividing 
by  the  product  O  x.  Oy.  O  z,  we  obtain — 

iii    

Ox     Oy      Oz 

which  quantities  are  represented  by  the  three  indices  in  the 
order  given  above.  Further,  if  we  consider  the  axes  x  andj' 
as  positive,  z  will  be  negative,  and  vice  versa ;  and  as  the 
intercept  of  the  latter  is  the  shortest,  it  will  have  the  largest 

FIG.  100. 


of  the  three  indices,  but  with  the  opposite  sign  to  that  of  the 
other  two  ;  so  that  th'e  general  symbol  becomes  [hkli],  in 


CHAP.  IV.]  Hexagonal  Notation.  77 

which  the  first  two  indices  are  independent,  the  third  being 
equal  to  their  sum,  but  with  the  opposite  sign.  The  number 
of  permutations  of  letters  and  signs  satisfying  these  con- 
ditions is  twelve,  where  order  is  shown  by  the  twelve 
divisions  numbered  like  a  clock-face,  making  up  the  interior 
polygon  in  the  diagram,  fig.  100.  They  represent  the  hori- 
zontal projection  of  the  faces  corresponding  to  any  value  of 
/,  and  when  this  is  greater  than  o  there  will  be  twelve  similar 
faces  below  the  horizontal  plane  having  /  in  their  symbols, 
the  whole  making  up  the  twenty-four-faced  solid  known  as  a 
dihexagonal  pyramid,  which  is  the  general  representative 
form  of  the  system.  These  symbols  are  given  in  full  on 
page  88. 

In  Weiss's  system,  the  lateral  axes  are  noted  as  a{  az  a3, 
their  positions  being  also  shown  in  fig.  100.  According  to 
this,  the  symbols  determining  a  face,  those  of  the  two  inde- 
pendent lateral  and  the  vertical  axes,  are  a  \  na  :  m  c,  while 
that  of  the  third  lateral  axis,  which  determines  its  position,  is 

fl  22. 

— — a  •  the  full  symbol  therefore  will  be  a  :  na  : a  :  me.1 

n—i  n—i 

the  last  term  referring  to  the  vertical  axis  ;  but  in  order 
to  express  the  relation  between  these  symbols  and  the 
Bravais  notation  it  is  necessary  to  invert  the  order  of  the 
first  three  and  change  the  sign  of  the  parameter  corre- 

72 

spending  to  /,  which  gives  the  form  a  :  n  a  :  a  :  m  c. 

7z— i 

subject  to  which  alteration  the  indices  in  [hklt]  will  be  the 
reciprocals  of  Weiss's  symbols.  Naumann's  contracted  sym- 
bol is  m  Pn,  which  is  of  analogous  signification  to  that  of 
the  hexakisoctahedron  in  the  cubic  system,  with  the  dif- 

1  A  notation  by  indices  corresponding  to  this  order,  of  the  form 
|  k  k  I,  in  which  h  and  k  refer  to  the  two  independent  lateral  axes  f , 
which  is  always  —h  —  k,  to  the  third,  and  /  to  the  vertical  axis,  is 
adopted  in  Groth's  treatise.  This  has  the  advantage  of  using  the 
index  letters  in  the  same  general  order  as  in  the  other  systems,  but  it 
has  not  been  adopted  in  any  subsequent  work. 


Systematic  Mineralogy, 


[CHAP.   IV. 


ference  that  P,  the  initial  of  pyramid,  is  substituted  for  O  as 
the  type-unit  form ;  m  is  the  parameter  coefficient  of  the 
vertical,  and  n  that  of  the  longer  lateral  axis. 

The  lateral  axes  are  related  to  the  vertical  in  some  con- 
stant ratio  expressed  by  a  :  c,  where  if  a  =  i,  c  is  always 
some  irrational  number  greater  or  less  than,  but  never  equal 
to,  unity.  This  is  known  as  the  fundamental,  or  unit-axial 
ratio,  and  the  form  from  which  it  is  derived  as  the  unit-form 
of  the  species,  as  in  all  other  forms  of  the  same  substance 
the  lengths  of  lateral  axes  are  expressed  as  rational  multiples 
or  submultiples  of  unity,  and  that  of  the  vertical  as  similar 
rational  modifications  of  c.  The  particular  ratio  is,  however, 
only  constant  for  the  crystals  of  the  same  substance  ;  and 
therefore,  unlike  the  cubic  system,  in  which  the  forms  consti- 
tute only  a  single  series,  there  are  as  many  series  of  hex- 
agonal forms  as  there  are  different  minerals  crystallising  in 
that  system,  which  is  also  true  for  all  the  remaining  systems. 

The  dihexagonal  pyramid  is  represented  in  perspective 


FIG.  101. 


FIG.  102. 


elevation  in  fig.  loi,1  and  in  horizontal  projection  fig.  102, 
the  first  being  noted  as  an  imaginary  form  {1233}  or  P  -2,  in 

1  This  is  intended  to  represent  the  effect  of  a  model  whose  faces 
are  a  mere  skin  covering  the  edges  of  the  planes  of  symmetry  which 
are  exposed  by  the  removal  of  the  middle  portion  of  the  covering.  The 
principal  plane  is  indicated  by  strong,  and  the  lateral  axial  planes 
by  Ught,  shading  ;  the  interaxial  planes  are  white. 


CHAP.  IV.]  Dikexagonal  Pyramid.  79 

which  a  \  c=  i  :  i  *i,  while  the  second  has  the  general 
notation,  the  faces  being  further  shaded  in  accordance  with 
the  scheme  of  triads  in  the  table  on  page  88.  Its  twenty- 
four  faces  meet  in  thirty-six  edges  of  three  different  kinds — 
namely,  twelve  middle  or  basal  edges  in  the  principal  plane 
of  symmetry,  and  twenty-four  terminal  or  polar  edges,  which 
lie  alternately  in  the  lateral,  axial,  and  interaxial  planes,  and 
meet  the  vertical  axis  in  a  twelve-faced  solid  angle  at  an 
equal  distance  on  either  side  of  the  basal  plane  j  their  other 
extremities  form  with  the  basal  edges  twelve  four-faced 
middle  or  basal  solid  angles.  The  dihedral  angles  over  the 
basal  edges  are  all  similar  ;  those  over  the  polar  edges  are 
alternately  larger  and  smaller,  but  never  equal,  as  that 
requires  the  basal  section  to  be  a  regular  dodecagon,1  which 
is  excluded  as  giving  for  n  the  irrational  value  v/2  sin.  75° 
=  1-3666.  .  .  .  For  any  value  of  n  lower  than  this,  the 
more  obtuse  polar  edges  lie  in  the  interaxial  planes,  and 
n  =  i,  their  angle  =180°,  or  the  twelve  faces  are  reduced 
to  six,  giving  the  normal  hexagonal  pyramid,  or  that  of  the 
first  order,  shown  in  elevation  in  fig.  io32  and  in  plan  in 
fig.  104,  and  in  the  outer  hexagonal  figure  of  fig.  100.  This 
is  contained  by  twelve  isosceles  triangles,  forming  a  six 
faced  pyramid  on  either  side  of  the  regular  hexagonal  base. 
The  dihedral  angle  over  a  basal  edge  is  equal  to  twice  the 
oblique  angle  at  the  base  of  a  right-angled  plane  triangle 
whose  perpendicular  and  base  are  respectively  the  vertical 

1  By  constructing  this  solid,  or,  what  is  easier,  projecting  it  on  its 
base,  it  will  be  seen  to  be  symmetrical  about  twelve  instead  of  six 
lateral  axes,  and  to  have  twelve-  instead  of  six-part  symmetry  about  its 
principal  axis,  or  to  have  higher  symmetry  than  that  defined  above  as 
characteristic  of  the  system.     This  illustrates  the  statement  on  page  9, 
that  solids  of  higher  than  senary  symmetry  are  excluded  by  the  prin- 
ciple of  rationality. 

2  The  edges  of  the  interaxial  planes  are  shown  as  dotted  lines,  to 
signify  that  these  are  no  longer  effective  edges  of  form,  but  only  of 
symmetry. 


So 


Systematic  Mineralogy. 


[CHAP.  IV. 


and  a  lateral  interaxis,  and  if  the  former  be  called  c,  the 
latter  a',  and  the  observed  basal  angle  ft,  we  have  : 


But  a'  is  inclined  at  30°  to  an  adjacent  lateral  axis  a  ; 
therefore  whefi  a  is  made  =i,  #'=!</  3  and  a  :  c=  i  : 


tan.  -,  which  corresponds  to  the  fundamental  ratio  of 
the  species,  if  the  particular  pyramid  measured  be  assumed 


FIG.   103. 


FIG.  104. 


as  the  unit  form  of  the  series.  The  same  quantity  may  also 
be  found  from  the  dihedral  angle  over  a  polar  edge  by  the 
expressions  : 

Sin.  £  =  cotan.  2   ^   ancj  tan>  £  _  m  ^  or  _  c  wnen 

m  =  i,  where  y  =  the  measured  angle  and  £  the  inclination 
of  that  edge  to  the  lateral  axis  lying  in  the  same  plane 
with  it. 

The  hexagonal  pyramid  is  the  particular  form  of  \h  k  //} , 
in  which  h  =  i  and  k  =  o,  whence  /  =  T.  If  /  also  =  i , 
the  general  symbol  becomes  {101  i},  or  that  of  a  face 
parallel  to  the  right  and  left  axis  k.  To  obtain  uniformity 
with  other  systems  it  is  customary,  however,  to  take  not 
this,  but  the  face  meeting  the  positive  pole  of  that  axis,  or 


CHAP.  IV.]       Pyramid  of  the  Second  Order. 


8l 


{o  i  i  i},  as  representative  of  the  form.  The  notation  of  the 
six  upper  faces,  in  which  z  is  positive,  is  given  in  figs.  104 
and  100.  Weiss's  symbol  is  cca  :  a  :  a  :  c,  and  Naumann's 
n  P  or  P  when  n  =  i  signifying  the  unit  form. 

When  n  in  the  dihexagonal  pyramid  is  greater  than 
1.366.  .  .  .,  the  more  obtuse  polar  edges  are  those  in  the 
lateral  axial  planes,  their  dihedral  angles  becoming  180° 
when  n  =  2.  This  gives  the  hexagonal  pyramid  of  the 
diagonal  position  or  second  order,  seen  in  elevation  in  fig. 
105  and  plan  in  fig.  106,  and  the  strong-lined  hexagon  in 
fig.  100.  It  has  the  same  general  geometrical  properties  as 


FIG.  105. 


FIG.  106. 


the  pyramid  of  the  first  order  from  which  it  differs  in  the 
position  and  relative  lengths  of  the  axes  ;  half  the  measured 
angle  across  a  basal  edge  gives  the  oblique  angle  at  the  base 
of  a  right-angled  triangle  whose  base  and  perpendicular  are 
in  the  ratio  of  a  :  c. 

Any  face  cuts  two  adjacent  positive  semi-axes  at  the 
same,  and  the  intermediate  negative  one  at  half  that  dis- 
tance or  h  =  k  =  i,  whence  7=  2.  The  general  symbol  is 
therefore  {i  i  2*'},  or  for  the  form  with  the  unit  vertical 
axis  {i  122},  corresponding  to  Weiss's  2  a  :  2  a  :  a  :  c. 
Naumann's  general  symbol  is  mP2  or  P2  for  the  unit 
form.  The  full  notation  of  the  upper  half  is  given  in  fig.  106. 

If  the  length  of  the  vertical  axis  in  either  class  of 
G 


32  Systematic  Mineralogy.  [CHAP.  IV. 

pyramid  be  varied  by  multiplying  its  unit-value  by  any 
rational  coefficient,  m,  which  may  be  either  greater  or  less 
than  unity,  other  pyramids  upon  the  same  base  will  be 
obtained  which  will  be  steeper,  or  their  basal  angle  will 
increase  proportionately  with  that  of  m,  and  vice  versci,  as  in 
fig.  107,  representing  three  normal  hexagonal  pyramids  of 
different  altitudes.  If  the  middle  one  of  these  be  regarded 


FIG.  107. 


FIG.  108. 


as  P or  {o  i  i  i},  the  outermost  will  be  2  POT  {0221},  and 
the  innermost  ^  P  or  {0112};  if  the  latter  is  considered  as 
P  the  others  will  be  2  P  and  4  P  or  {0441};  and  lastly,  if 
the  outermost  is  the  unit,  the  others  will  be  %  P  and  £  P  or 
{o  1 1 4}  respectively.  When  m  =  ao  or  i  =  o  the  basal 
angles  become  180°,  and  as  those  of  the  polar  edges  change 
in  the  reverse  order  they  diminish  until  they  correspond  to 
the  plane  angle  of  the  base  ;  or,  in  other  words,  the  pyramid 
becomes  a  prism.  The  forms  of  this  class  corresponding  to 
the  three  kinds  of  pyramids  are  represented  in  figs.  108,  109, 
and  no.  The  first  of  these,  derived  from  fig.  101,  is  a 
dihexagonal  prism  contained  by  twelve  similar  rectangular 
faces  meeting  in  edges  parallel  to  the  vertical  axes  at  angles 
which  are  alternately  greater  and  less  than  150°  (that  being 
the  angle  of  a  regular  dodecagon,  which  is  excluded  by  the 


CHAP.  IV.] 


Hexagonal  Prism. 


irrationality  of  its  parameters).     The  symbols,  as  will  be 
apparent  from  the  derivation,  are 

_          n 
[h  k  I  o} ,    _     a  :  n  a  :  —  a  :  co  c,  and  GO  Fn. 

Fig.  109  is  the  hexagonal  prism  of  the  first  order,  con- 
tained by  six  faces  meeting  at  120°  in   the  lateral   axial 


FIG.  iog. 


FIG.  no. 


planes,  or  its  base  is  a  regular  hexagon  similar  in  position 
to  that  of  the  unit-pyramid.     It  is  represented  by 

{o  i  i  o},  GO  a  :  a  :  —a:<x>c,  and  co  P. 

Fig.  no  is  the  hexagonal  prism  of  the  second  order, 
derived  from  the  corresponding  pyramid,  fig.  105.  It  is 
also  on  a  regular  hexagonal  base,  its  edges  lying  in  the 
lateral  interaxial  planes,  and  is  represented  by 

{1120},  2  a  :  2  a  :  —a  :  GO  c,  and  <x>P 2. 

As  all  the  faces  of  prisms  lie  in  single  zones  which  are 
unlimited  in  the  direction  of  their  axes,  they  cannot  of  them- 
selves form  complete  crystals,  but  can  only  appear  in  com- 
bination. These  are  said  to  be  open  forms,  and  are  common 
in  all  except  the  cubic  systems,  where  the  simple  forms  are 
necessarily  closed. 

When  m  is  less  than  i,  the  basal  edges  of  the  pyramid 
become  sharper  and  the  polar  ones  blunter  than  those  of 

G  2 


84  Systematic  Mineralogy.  [CHAP.  IV. 

the  unit  form  ;  and  when  ///  =  o  the  angles  of  the  former 
are  o°  and  of  the  latter  180°,  or  the  whole  of  the  faces 
meeting  the  vertical  axes  at  either  extremity  fall  into  the 
same  surface,  and  the  form  is  reduced  to  the  single  unlimited 
plane  containing  the  lateral  axes.  When,  however,  this 
is  shifted  from  the  central  position,  it  will  intercept  some 
length  upon  the  vertical  axis,  and  require  a  corresponding 
parallel  face  on  the  opposite  side  of  the  origin  ;  the  com- 
plete form  is  therefore  represented  by  two  faces  parallel  to 
the  basal  section.  This,  known  as  the  basal  or  terminal 
pinakoid,  is  another  form  only  possible  in  combination, 
having  no  proper  geometrical  form,  its  shape  being  con- 
ditioned by  the  edges  formed  in  combination.  For  instance, 
in  fig.  108,  it  is  dihexagonal,  in  fig.  109  hexagonal  of  the 
first  order,  and  in  fig.  no  of  the  second,  these  differences 
being  obviously  due  to  the  different  prisms  with  which  it  is 
combined.  The  symbols  are  : 

{oooi},  ooa  :  <x>a  :  <x>a  :  c,  and  oP. 

The  seven  classes  of  forms  described  above — namely, 
the  three  pyramids,  with  their  corresponding  prisms  and  the 

„  terminal  pinakoid,  are  the  only  kinds 

oP      oP        oP  .,,       ..    f  ,, ' 

possible  with  full  hexagonal  symmetry 

z          to  seven  axes,  as  will  easily  be  seen  by 
'"m^2  gi^g  special  values  to  the  indices  in 
the  general  symbol  h  k  Fi,  and  working 

Pz  them   out    subject    to    the   condition 

h  +  k  +  /  =  o.    Their  relation  to  each 
mP...mPn...mP2  otner  js  ^est  seen  m  Naumann's  dia- 

^  p  2  gram,  formed  byarranging  their  symbols 
as  in  the  margin.  The  first  vertical  line 
contains  the  hexagonal  pyramids  of  the  first  order;  the 
second,  the  dihexagonal  pyramids;  and  the  third,  the  pyramids. 
of  the  second  order.  The  unit  forms  of  these  three  classes 
are  in  the  third  horizontal  line;  their  obtuse  modifications 


CHAP.  IV.]          Hexagonal  Combinations.  85 

having   m  <  i,  or  generally  a  proper  fraction  expressed 

as  —  in  the  second :  the  fourth  line  contains  the  more  acute 
m 

pyramids  having  m  >  i ;  the  fifth,  the  corresponding  infinitely 
acute  form  or  prisms,  and  the  top  line  the  common  limit  of 
obtuse  form,  or  the  terminal  pinakoid.  Any  one  of  these 
forms  lies  in  the  same  zone  with  any  other  in  the  same  line, 
whether  horizontal  or  vertical.  In  the  use  of  this  scheme  it 
must  always  be  remembered  that  the  rational  coefficient  ;// 
applies  not  to  the  natural  unit  number  i,  but  to  the  funda- 
mental arbitrary  ratio  a  :  c.  From  what  has  been  said  con- 
cerning the  properties  of  pyramids  of  varying  altitudes,  it  will 
be  easily  seen  that  the  basal  angles  vary  more  rapidly  than 
those  over  the  polar  edges,  as  they  may  range  from  o°  to  180°, 
while  the  latter  can  only  vary  between  120°  and  180°.  In 
the  description  of  hexagonal  minerals,  therefore,  the  angle  of 
the  basal  edge  is  usually  given  as  that  characteristic  of  the 
species,  and  determining  the  ratios  a  :  c  most  accurately. 


FIG. 


FIG.  112. 


In  combinations  of  hexagonal  forms  of  the  same  order, 
the  more  acute,  or  those  with  the  highest  m  or  lowest  /',  bevel 
the  basal  edges  of  the  more  obtuse,  and,  conversely,  the  latter 
acuminate  or  blunt  the  polar  summits  of  the  former.  The 

*G3 


86  Systematic  Mineralogy.  [CHAP.  IV. 

steepest  form  of  any  series,  the  prism,  and  the  flattest,  the 
terminal  pinakoid,  respectively  truncate  the  basal  edges  and 
polar  summits  of  any  pyramid,  as  in  fig.jn,  containing 
ooP,  3P,P,oP,or  {oiio},  {0331},  {ono},  {oooi}. 


FIG.  113. 


FIG.  114. 


In  combinations  of  forms  of  different  orders,  having  m 
in  common,  or  of  the  same  altitude  upon  different  bases, 


FIG.  115. 


FIG.  116. 


\  I 


m  P  2  truncates  the  polar  edges  of  m  P,  and  the  latter  bevels 
the  basal  solid  angles  of  the  former,  as  in  figs.  112,  113, 
while  mP  truncates  the  edges  of  mPn  lying  in  the  lateral 


CHAP.  IV.] 


Hexagonal  Combinations. 


FIG.  117. 


interaxial  planes  (figs.  114,*  115),  and  mP-z,  those  in  the 
lateral  axial  planes,  the  relations  of  the  three  classes  of 
prisms  being  similar  to  those  of  the  corresponding  pyramids. 
In  combinations  of  forms  in  which  both  m  and  n,  or  base 
and  altitude,  are  different,  mPn,  when  combined  with  a 
more  acute  hexagonal  pyra- 
mid of  either  order,  appears 
as  an  obtuse  twelve-faced 
point  upon  the  polar  summit 
of  the  latter,  as  seen  in  plan 

(fig.  1 1 6),  for  P-  Pn.    A 
in 

pyramid  of  the  second 
order  upon  a  more  acute 
form  of  the  first  forms  an  an- 
alogous six-faced  point  (fig. 
117),  the  plan  of  n  P,  Pz. 
In  the  reverse  condition,  when  the  pyramid  of  the  second 


FIG.  118. 


FIG.  119. 


order  is  the  more  acute  form,  its   faces  modify  the   solid 
angles  in  the  basal  section  of  that  of  the  first  order,  the 

1   This  is  noted  as  P  P*,  or   (o  1 1  ij    {1233}. 


88  Systematic  Mineralogy.  [CHAP.  IV. 

aspect  of  such  combinations  varying  with  the  differences  in 
the  parameters  of  the  two_forms.  For  instance,  in  fig.  118 
the  faces  of  2  Pz  or  {i  i  2  1}  make  new  edges,  which  are 
parallel  to  the  polar  edges  of  P  or  {o  1 1 1}  ;  while  in  fig.  1 19 
the  combination  edges  are  parallel  to  the  polar  ones  of  the 
replacing  form  '  f  P  2,  or  (2  2  4  3} .  Lastly,  when  m  =  GO,  the 
basal  solid  angles  of  the  less  acute  form  are  truncated,  or, 
in  other  words,  the  prism  of  either  order  truncates  the  basal 
solid  angles  of  the  pyramid  of 
the  other,  as  in  fig.  120,  which  is 
noted  as  P,  <x>P  2,  o  P,  but  would 
do  equally  well  for  P2,  GO  P  o  P, 
if  the  order  of  the  first  two  con- 
stituents be  supposed  to  be  re- 
versed. The  above  include  all 
the  simpler  cases  of  holohedral 
hexagonal  combinations,  but  they 
are  not  very  commonly  met  with  in  nature,  there  being 
but  few  species  in  which  full  hexagonal  symmetry  pre- 
vails, and  these  are  generally  remarkable  for  the  large 
number  of  forms  present,  the  most  of  which  are,  however, 
as  a  rule,  very  subordinate  to  the  dominant  form,  gene- 
rally a  prism.  A  few  examples  of  such  combinations, 
which  are  usually  best  represented  in  horizontal  pro- 
jections, will  be  found  in  the  volume  on  'Descriptive 
Mineralogy.' 

Hemihedral  hexagonal  forms.  The  faces  of  the  di- 
hexagonal  pyramid  divide  symmetrically  into  four  groups  of 
six,  which,  as  the  corresponding  faces  above  and  below  the 
basal  section  differ  only  in  the  sign  of  their  fourth  index, 
may  be  represented  by  the  four  triads  indicated  by  dif- 
ferent shadings  in  the  horizontal  projection  (fig.  102),  cor- 
responding to  the  following  table,  in  which  the  faces  are 

1  In  other  words  P  truncates  the  polar  edges  of  |  P2,  a  relation 
which  is  very  commonly  observed  in  natural  crystals. 


CHAP.  IV.]  Hexagonal  Hemihedrism. 


89 


numbered  in  order  from  left  to  right,  commencing  on  the 
right  side  of  the  positive  semi-axis  h  : — 


A 

B 

C 

D 

i.  Ihki 
v.  klJi  i 

n.  khli 
vi.  Ik  hi 

III.    /Z/£/z" 

vn.  Ihki 

iv.  ^  /  k  / 

VIII.    >?^// 

ix.  Jikli 

x.  hlki 

XI.    £/^2 

xii.  Ik  hi 

E 

F 

G 

H 

xni.  Ihkl 

xiv.  /£  h  1  1 

xv.  h  k  1  1 

xvi.  hlki 

xvii.  klhl 

XVIII.    //£^Z 

xix.  //;/£? 

xx.  khll 

xxi.  hkll 

xxn.  hlki 

xxin.  klhl 

XXIV.    /I^Z 

This  may  be  halved  symmetrically  in  the  three  following 
ways,  each  corresponding  to  a  possible  case  of  hemihedrism 


A    C 
F     H 


and 


and 


and 


B     D 
E    G 


C    D 

E     F 


In  the  first  case,  one  form  contains  uneven  numbered 
faces  above  alternating  with  even  numbered  ones  below,  and 
the  other  even  ones  above  and  uneven  below.  This  is  known 
as  flagihedral  or  trapezohedral  hemihedrism  ;  in  the  second, 
or  rhombohedral  hemihedrism,  the  grouping  is  by  alternate 
pairs  of  faces  above  and  below  the  basal  section,  or  the  ist, 


Systematic  Mineralogy. 


[CHAP.   IV. 


3rd,  and  5th,  with  the  8th,  loth,  and  i2th  pairs  ;  and  the 
2nd,  4th,  and  6th  with  the  7th,  pth,  and  nth  pairs.  In  the 
third,  or  pyramidal  hemihedrism,  the  grouping  is  symmetrical 
to  the  base,  one  form  containing  all  even  and  the  other  all 
odd  numbered  faces. 

Of  the  two  most  obvious  methods  of  dividing  the  table 
into  upper  and  lower,  and  right  and  left  halves,  the  first  is 
excluded  by  not  satisfying  the  conditions  of  symmetry,  giving 
forms  having  the  indices  of  the  vertical  axis  either  all  posi- 
tive or  all  negative.  The  second,  a  one-sided  distribution 
of  faces,  has  been  called  trigonotype  hemihedrism  by  some 
writers,  while  others  say  that  it  does  not  possess  true  hemi- 
hedrism. This  is  possibly  rather  a  question  of  terms  than  of 
fact.  It  need  not  be  discussed,  as  the  form  does  not  occur 
in  minerals. 

Trapezohedral  hemihedrism.  The  dihexagonal  pyramid, 
when  divided  hemihedrally  in  the  manner  shown  in  fig.  122, 
corresponding  to  the  first  case,  as  defined  above,  gives  rise 
to  the  two  hemihedral  forms  (figs.  121  and  123),  the  first 


FIG.  121. 


FIG.  122. 


FIG.  123. 


from  the  white,  and  the  second  from  the  shaded  faces  of  fig. 
122.  Fig.  124  is  a  horizontal  of  projection  of  fig.  123  ;  the 
faces  are  numbered  according  to  the  table  on  page  89,  the 
edges  below  the  horizontal  plane  being  shown  in  dotted 
lines.  These  known  as  hexagonal  trapezohedra  are  con- 


CHAP.  IV.]        Trapezoftedral  Hemihedrism.  91 

tained  by  twelve  trapeziform  faces  meeting  in  twelve  similar 
polar  and  twelve  dissimilar  middle  edges,  the  latter  being 
alternately  longer  and  shorter.  Like 
the  plagihedral  hemihedra  of  the  cubic 
system,  they  have  the  same  number 
and  kind  of  axes  of  symmetry  as  the 
holohedral  forms,  but  no  planes  of  sym- 
metry, and  are  therefore  non-super- 
posable. 

As  a  face  of  a  hexagonal  pyramid 
of  either  order  or  a  dihexagonal  prism 
contains  two,  that  of  a  hexagonal  prism  four,  and  the  ter- 
minal pinakoid  six,  faces  of  the  general  form  [hklt],  it  is 
clear  that  none  of  these  holohedral  forms  will  be  geometri- 
cally  changed  by  this  kind  of  hemihedry,  which  is  therefore 
only  effective  in  producing  new  forms  in  the  dihexagonal 
pyramid.  No  examples  of  crystals  of  this  kind  of  hemihe- 
drism, whether  of  natural  or  artificial  origin,  are  known,  so 
that  as  yet  they  only  represent  a  geometrical  possibility, 
^t  may  be  represented  by  the  symbols 


K"  {hkli},\ 


:  na  ;  —a  :  me 


and 


mPn        m Pn  , 


Rhombohedral  hemihedrism.  The  second  method  of  hemi- 
hedral  division  of  the  dihexagonal  pyramid,  that  by  alternate 
pairs  effaces  above  and  below  the  base,  as  in  fig.  126,  pro- 
duces from  the  white  faces  fig.  125,  and  from  the  shaded 
ones  fig.  127.  The  corresponding  horizontal  projections  are 
seen  in  figs.  128  and  129.  A  form  of  this  class,  known  as  a 

1  The  hemihedrism  is  here  shown  by  the  prefix  \  to  the  holohedral 
symbol  without  special  indication  of  the  particular  case  meant,  and  is 
therefore  general  for  all  kinds. 


Systematic  Mineralogy. 


[CHAP.  IV. 


scalenohedron,  a  contraction  of  scalene  dodecahedron,  is  con- 
tained by  twelve  faces,  all  similar  scalene  triangles  meeting 


FIG.  125. 


FIG.  126. 


FIG.  127. 


in  two  kinds  of  polar  edges,  six  longer  or  more  obtuse  (x),  six 
shorter  or  more  acute  (Y),  and  six  middle  edges  (z),  lying  in 
zigzag  order  about  the  basal  section. 

From  the  derivation  of  this  form  it  will  be  apparent  that 


FIG.  128. 


FIG.  129. 


the  longer  polar  edges  have  the  same  character  as  in  the 
holohedral  form,  and  that  new  shorter  ones  (Y)  lie  in  the 
same  transverse  sections  with  them — that  is,  in  the  lateral 


CHAP.  IV.]         RJiomboJiedral  Hemihedrism.  93 

interaxial  planes,  which  are  therefore  planes  of  symmetry, 
while  the  lateral  axial  planes  are  not  ;  and  from  the  obliquity 
of  the  middle  edges  (z)  to  the  original  basal  section,  the 
latter  cannot  be  a  plane  of  symmetry.  The  lateral  crystallo- 
graphic  axes  being  normals  to  the  interaxial  planes,  they  will 
be  axes  of  binary  symmetry,  and  as  these  planes  also  inter- 
sect in  the  vertical  axes,  the  latter  will  also  be  an  axis  of 
symmetry,  but  reduced  from  hexagonal  to  ternary.  We 
therefore  have  ternary  symmetry  about  the  principal  axis, 
binary  about  the  lateral  crystallographic  axes,  and  three 
planes  of  symmetry  inclined  at  60°  to  each  other  as  charac- 
teristics of  this  class  of  hemihedrism. 

The  two  scalenohedra  derivable  from  the  same  di- 
hexagonal  pyramid  are  superposable  —  that  is,  either  may  be 
brought  into  coincidence  with  the  other  by  rotation  through 
60°  or  1  80°  about  the  vertical  axis.  They  are  distinguished 
as  positive  and  negative,  or  direct  and  inverse  forms,  accord- 
ing to  position.  The  choice  of  position  is,  however,  arbi- 
trary, and  usually  depends  upon  structural  peculiarities. 

The  general  symbols  of  the  scalenohedron  are: 


(7  /  T  -i       j     ,  mPn 

K  \hkli\  and   +  -  ,  _  --- 

2  2 

The  same  selection,  applied  to  the  hexagonal  pyramid  of 
the  first  order,  as  in  fig.  131,  produces  the  two  hemihedral 
forms,  figs.  130  and  132  —  the  former  from  the  white  and 
the  latter  from  the  shaded  faces.  Figs.  133  and  134  are  the 
corresponding  horizontal  projections.  A  form  of  this  class, 
known  as  a  rhombohedron,  a  contraction  for  rhombic 
hexahedron,  is  contained  by  six  faces,  all  equal  rhombs, 
meeting  in  three  polar  edges  at  either  end  of  the  vertical 
axis,  and  six  middle  edges  in  zy  order  about  the  basal 
section.  The  polar  diagonals  of  the  faces  meeting  the 
vertical  axis,  as  shown  in  the  dotted  line  in  fig.  130,  represent 
the  obtuse  edges  (x)  of  the  scalenohedron,  as  will  be  readily 
seen  when  it  is  remembered  that  the  hexagonal  pyramid  is 


94  Systematic  Mineralogy.  [CHAP.  IV. 

that  particular  dihexagonal  pyramid  whose   faces  meet  at 
1 80°  in  the  obtuse  polar  edges.     The  symmetrical  relations 


FIG.  130. 


FIG.  131. 


FIG.  132. 


are  therefore  in  every  respect  similar  to  those  of  the  scaleno- 
hedron.    The  rhombohedron,  being  the  most  important  form 


FIG.  133. 


FIG.  134. 


of  its  class,  and  even  of  its  system,  is  considered  as  cha- 
racteristic of  this  kind  of  hemihedrism,  which  is  therefore 
called  rhombohedral  instead  of  scalenohedral. 

The  symbols  of  the  two  rhombohedra  derived  from  the 
same  (unit)  form  are  : — 

P      P 
*.-{QI  i  I},K  {101  i},and  +  -  —  —. 

2  2 

The  dihedral  angles  over  the  two  kinds  of  edges  in  the 
rhombohedron  are  mutually  supplemental,  er  one  kind  is 
as  much  above  as  the  other  is  below  90°.  When  the  larger 
angle  is  in  the  polar  edges,  the  rhombohedron  is  obtuse,  but 


CHAP.  IV.]  Rhomboliedm.  95 

when  it  is  in  the  middle  edges  it  is  acute.  The  form  occu- 
pying the  middle  position,  or  having  the  angles  of  both 
polar  and  middle  edges  right  angles  is  the  cube,  which  is  a 
possible  rhombohedron,  deriving  from  the  hexagonal  pyra- 
mid, having  a  :  c==  i  :  1*2247  j  and  although  it  is  not  known 
to  exist  in  nature,  there  are  several  forms  very  nearly 
approaching  it.  This  is  an  example  of  what  are  called 
limiting  forms,  where  the  same  geometrical  solid  may  arise 
in  two  systems.  In  this  case  it  is  obviously  possible,  from 
the  circumstance  that  the  ter-quaternary  symmetry  of  the 
cube  includes  the  lower  ter-binary  kind  of  the  rhombohedron, 
and  the  real  test  of  the  nature  of  the  form  is  to  be  looked 
for  not  in  the  presence  of  the  lower;  but  the  absence  of  the 
higher  symmetry,  which  is  usually  apparent  in  the  character 
of  its  combinations  ;  but  assuming  it  to  appear  as  a  simple 
form,  its  true  nature  could  only  be  determined  by  optical 
or  other  investigation  of  the  structural  peculiarities  of  the 
substance. 

The  symbols  of  the  two  positive  and  negative,  or  direct 
and  inverse,  rhombohedra,  originating  from  the  same  hexa- 
gonal pyramid,  where  the  latter  is  a  unit  form  : 

P*      P 

K  (o  i  i  i},  K  [i  o  i  i},  and  +—  .   — -  ; 

2  2 

or,  generally, 

,       -  .,                mP         mP 
K{O  :  i  z),  K  {i  o  i  z),  and  -\ 

Like  the  scalenohedra  they  are  superposable,  either  one 
being  brought  into  coincidence  with  the  other  by  rotation 
through  60°  or  180°  about  the  vertical  axis.  From  their 
derivation  it  will  be  apparent  that  in  combination  they 
truncate  each  other's  solid  angles  obliquely,  as  in  fig.  135  ; 
and  when  the  two  are  exactly  balanced,  the  hexagonal 
pyramid  (fig.  131)  is  reproduced. 

A  rhombohedron  of  either  position  and  any  length  of 
vertical  axis  has  its  polar  edges  truncated  by  the  faces  of 


96  Systematic  Mineralogy.  [CHAP.  IV. 

another  of  the  same  series  of  the  opposite  position,  whose 
height  is  one  half  or  breadth  of  base  twice  that  of  the  first, 


FIG.  135. 


as  in  fig.  136.     This  case  is  commonly  observed  in  calcite, 

p 
whose   principal  rhombohedron    +  -    has  its  polar  edges 

—  P 

replaced  by  faces  of  —  ?—  ,  and  similarly  modifies  those  of 


~  P 

*  —  > 

2 


2  P  —  P 

—  _  .  and  in  the  same  way  —  %_  is   modified   by 

2  2 

-L?by  +!/*,  and  so  forth. 

2  2 

By  varying  the  value  of  m  in  the  symbol  of  a  rhombo- 
hedron, a  series  of  other  forms  of  greater  or  less  altitude 
upon  the  same  base  are  obtained  in  the  same  way  as  with 
the  pyramid.  As  m  decreases,  the  angles  of  the  middle  edges, 
as  well  as  their  obliquity  to  the  basal  section,  diminish  as 
those  of  the  polar  edges  increase^  the  latter  becoming  180° 
when  m  =  o,  producing  the  basal  pinakoid.  In  the  opposite 
direction  the  middle  edges  become  more  obtuse  and 
increase  their  inclination  to  the  basal  plane  ;  and  when 
m  —  GO  they  fall  into  the  same  vertical  lines  with  the  polar 
edges,  or  the  hexagonal  prism  GO  .Pis  produced.  The  limits 
of  the  series  of  the  rhombohedron  are  therefore  the  same 
as  those  of  the  hexagonal  pyramid  of  the  first  order  upon  the 
same  base. 

The    same  kind  of  relation  subsists    in   one  direction 


CHAP,  iv.]     Rhombokedra  and  ScalenoJiedra. 


97 


between  the   scalenohedron  and  its  holohedral  form,  and 


•  • 

therefore,  by  increasing  the  value  of  m  in  -  ,  more  acute 

2 

forms  upon  the  same  base  may  be  obtained  up  to  a  dihexa- 
gonal  prism  <x>  P  n  •  but  when  m  is  diminished,  the  obtuse 
dihedral  angle  of  the  edges  (x)  increases  more  rapidly  than 
that  of  the  acute  ones  (Y)  (fig.  125),  and  becomes  180°  when 
the  latter  has  still  a  measurable  inclination,  while  that  of 
the  middle  ones  is  unchanged.  The  twelve  faces  at  either 
end  are  therefore  changed  into  six,  producing  a  rhombo- 
hedron,  when  the  value  of  m  is  still  far  in  excess  of  o  ;  or 
the  inferior  limit  of  the  series  of  the  scalenohedron  is  not 
the  basal  pinakoid,  but  a  rhombohedron  whose  symbol  is 

m(2  —  n)  r> 

\  _  if  mf>ti 

n         ,  when  that  of  the  scalenohedron  is  _  _  .  As 

2 

2 

these  forms  have  their  middle  edges  in  common,  the  rhom- 
bohedron is  completely  enclosed  in  the  scalenohedron,  as 


FIG.  137. 


FIG.  138. 


in  fig.  137.     This  is  known  as  the  rhombohedron  of  the 
middle  edges. 


98  Systematic  Mineralogy.  [CHAP.  IV. 

There  are  two  other  rhombohedra  included  in  any 
scalenohedron,  each  having  its  polar  edges  parallel  to  one 
or  other  kind  of  the  same  edges  in  the  latter  form.  The 
first  of  these,  or  rhombohedron  of  the  shorter  polar  edges 
(fig.  138),  is  similar  in  direction  to  the  scalenohedron,  and  is 

m(2n  —.1)^0 
represented  by  the  symbol          n  _  ,  while  the  second,  or 

2 

rhombohedron  of  the  longer  polar  edges  (fig.  139),  which  is 
the  most  acute  one  that  can  be  included,  is  inverse  in 

*»(«+*)ip  p 

direction,  having  the  symbol  —        n          '  when  ~  _   is 


2 


positive,  and  vice  versA.  In  this  the  length  of  the  "vertical 
axis  is  the  sum  of  those  of  the  other  two,  as  will  be  seen  by 
comparing  the  factors  measuring  this  dimension  in  the  three 
symbols,  as  — 

n  +  i  =  (2  n  —  i)  +  (2  —  n). 

The  special  values  of  these  three  kinds  of  rhombohedra 
for  the  most  commonly  observed  scalenohedra  are  as  follows  : 


Scalenohedra $     • 

2 

Rhombohedra  of  middle  edges      .    - 

2'        2          2  -        2 

Rhombohedra  of  shorter  polar  edges  — ,    ^  ^     ^^        7^* 

22  2  '  2 

Rhombohedra  of  longer  polar  edges  —  ^~}  _  1^^  _  9_^  _  II^> 

2  2  2  '  2 

In  the  combinations  of  these  four  correlated  forms,  the 
most  obtuse  one,  the  rhombohedron  of  the  middle  edges, 
modifies  the  polar  summits  of  the  scalenohedron,  producing 
blunt  three-faced  points,  the  new  edges  being  parallel  to  the 
original  middle  edges,  as  in  fig.  140,  which  represents  the 


CHAP.  IV.]        ScalenoJiedral  Combinations. 


99 


common  scalenohedron  of  calcite,  reduced  by  cleaving  away 
its  points,  the  faces  produced  by  cleavage  being  those  of 
the  unit  rhombohedron  of  the  species.  The  rhombohedron 
of  the  shorter  polar  edges  truncates  the  longer  ones  of  the 
scalenohedron  of  the  same  direction  ;  and  that  of  the  longer 


FIG.  140. 


FIG.  141. 


FIG.  142. 


polar  edges  the  shorter  ones  of  the  inverse  scalenohedron,  as 

r   p 

in  fig.  141,  where  the  faces  of  rrr  will  be  those  of  —  °— ,  if 

the  scalenohedron  is  considered  as  + 

2 

The  pyramid  of  the  second  order,  the  three  classes  of 
prisms,  and  the  basal  pinakoid  are  not  changed  in  appearance 
by  rhombohedral  hemihedrism. 

In  combination,  the  prism  of  the  first  order  truncates 
the  middle  solid  angles  of  rhombohedra  and  scalenohedra 

alike  ;  in  the  first  case  the  new  faces  are  triangular  planes, 

p 
as   in   -.00 /'(fig.  142),  and  in  the  second  deltoidal,  as  in 

2 

f-.  oo/*  (fig.  143).      When,  however,  the  prismatic  edges 

proper,  those  parallel  to  the  vertical  axis,  are  apparent,  both 
prism    and    rhombohedron    appear   as    five-sided    figures 


ioo  Systematic  Mineralogy.  [CHAP.  iv. 

(fig.  144),  representing  one  of  the  commonest  kinds  of  Calcite 
crystals,  oo  p.  —  "*_.   The  prism  of  the  second  order  being 

2 

the  common  limit  of  both  rhombohedra  and  scalenohedra, 


FIG.  143. 


FIG.  144. 


FIG.  145. 


£010 


it  will,  in  combination,  truncate  their  middle  edges,  as  in 

•i  P3.  P 

i__5.«p/>2  (fig.  145),  and      .  00/^2  (fig.  146),  both  being 

2  2 

observed  cases  in  Calcite. 

The  basal  pinakoid  truncates  the  polar  summits  both  of 
FIG.  146.  scalenohedra  and  rhombohedra, 

producing  either  six-  or  three- 
sided  faces ;  the  latter  case  is 
illustrated  in  fig.  136. 

When  many  rhombohedra  and 
scalenohedra  of  different  values 
are  combined  together,  the  result- 
ing solid  is  often  of  great  com- 
plexity, and  its  proper  symmetry 
is  not  always  easily  seen.  In  such  cases  projections  of  the 


CHAP.  IV.]  Naiimanris  Notation.  101 

faces  upon  the  plane  of  the  base  are  very  useful,  and  are 
generally  to  be  preferred  to  perspective  figures. 

Naumanris  rhombohedral  notation.  As  by  far  the  larger 
number  of  hexagonal  minerals  are  rhombohedrally  hemi- 
hedral,  it  is  generally  convenient  to  adopt  the  rhombohedron 
as  the  unit  form  of  the  series  rather  than  the  pyramid,  which 
is  less  common,  and  as  a  rule  has  its  two  kinds  of  faces  so 
unequally  developed  as  to  be  more  properly  regarded  as  a 

combination  of  two  rhombohedra.    It  is  therefore  customary 

p 
to  write  the  symbol  -  as  +  J?,  which  represents  the  two 

2 

rhombohedra  deducible  from  any  particular  value  of  the 
ratio  a  :  c.  This  may  also  be  derived  from  the  angle  over  a 
polar  edge  in  a  rhombohedron,  or  from  the  supplement  of 
that  over  a  middle  edge,  by  the  expression 

cos.  i  r 

cos.  a  =  - — -2— , 
sin.  60° 

where  r  =  the  measured  angle,  a  the  side  of  a  spherical 
triangle,  corresponding  to  the  angle  at  the  vertex  of  the 
right-angled  plane  triangle  whose  perpendicular  and  base 
are  the  vertical  axis,  and  a  lateral  interaxis,  or  c  and  a! 
respectively,  whence 

—.  =  cotan.  a. 


But  as  the  lateral  axes  and  interaxes  a  :  a'  are  in  the  pro- 
portion of  i  :  ^\Xj,  when  a  —  i,  the  required  length  of  the 
vertical  axis  will  be 

£•  =  1^/3  cotan.  a. 

From  any  rhombohedron  ±  m  J?  by  altering  the  value  m, 
making  the  vertical  axis  longer  or  shorter,  a  series  of  forms 
of  the  same  kind  are  obtained,  ranging  from  the  basal 
pinakoid  o  R  to  the  prism  GO  R,  which,  when  arranged  in 


IO2  Systematic  Mineralogy.  [CHAP.  IV. 

order,  give  a  series  analogous  to  that  of  the  pyramid  on 
page  83,  or 

oR  ...  +  ~R  ...  +  R  ...  mR  ...  oo^?. 

~~  m 

The  most  obtuse  form  of  scalenohedron  being  the  rhombo- 
hedron  having  the  same  middle  edges,  the  symbol  of  the 
latter  will  serve  to  indicate  any  scalenohedron  upon  the 
same  base,  if  another  sign  be  added,  marking  the  number  of 
times  that  its  unit  vertical  axis  is  lengthened.  This  class  of 
symbol  has  the  following  forms  :  — 

',  or  ± 


in  which  m  refers  to  the  vertical  axis  of  the  rhombohedron 
of  the  middle  edges,  and  n  to  the  same  axis  in  the  corre- 
sponding scalenohedron.  These  lengths  are,  however,  re- 

lated to  each  other  in  the  constant  ratio  of  i  :  —  —  ,  and 

2  —  n 

therefore  the  rhombohedral  symbol  of  any  scalenohedron 

whose  dihexagonal  notation  is  ^^  will  be  m(2~  n">R  1L, 

2  n  2n 

2  n' 

and  conversely,  m'  R  n'  will  be  m'  n'  P  -  . 

»'  +  i 

In  this   notation  the    rhombohedra  enclosed    in  any 
scalenohedron,  mRn,  are  —  i.  That  of  the  middle  edges, 

;  2.  That  of  the  shorter  polar  edges,  __  (3  n  —  i)R  ; 


2 


and  3.  That  of  the  longer  polar  edges,  —  -($n  + 

2 

and  the  table  on  page  98  becomes  — 

Scalenohedra  ....  ^32^?!      3-K  2 

Rhombohedra  of  middle)          D 

,  JK          2  Ji  T.  K  4  Ji 

edges   .....   ) 
Rhombohedra  of  shorter)  D 

polar  edges   .     .     .   I      4*  7* 

Rhombohedra  of  longer), 
polar  edges   .     .     .   ) 


CHAP.  IV.]          Pyramidal  Hemihedrism.  103 

Pyramidal  hemihedrism.  This,  the  third  case  on  page 
89,  corresponds  to  the  extension  of  alternate  faces  in  the 
dihexagonal  pyramid,  to  the  obliteration  of  the  adjacent 
ones  on  the  same  side  of  the  basal  section,  half  the  original 
edges  in  that  plane  being  retained,  but  no  others.  The 
result  is  a  regular  hexagonal  pyramid,  indistinguishable  geo- 
metrically from  the  holohedral  forms,  but  differing  from 
them  in  the  position  of  the  polar  edges,  which  do  not  lie  in 


FIG.  147. 


FIG 


FIG.  149. 


the  same  planes  with  the  lateral  axes  or  interaxes,  but  in 
some  intermediate  unsymmetrical  position.  Fig.  147  is  the 
form  produced  from  the  left-hand  (white  or  even-numbered) 


FIG.  151. 


faces  in  fig.  148,  fig.  149  its  horizontal  projection,  and  fig. 
150  that  of  the  right-hand  form,  from  which  it  will  be 
seen  that  in  the  first  case  the  greatest  length  of  the  basal 
edges  is  to  the  left,  and  in  the  second  to  the  right  of  the 


IO4  Systematic  Mineralogy.  [CHAP.  IV. 

lateral  axes.  These  are  called  hexagonal  pyramids  of  the 
third  order  ;  they  have  one  axis  and  one  plane  of  hexagonal 
symmetry,  but  the  original  binary  symmetry  is  completely 
lost  By  a  similar  kind  of  derivation,  prisms  of  the  third 
order  are  derived  from  the  dihexagonal  prism,  but  none  of 
the  remaining  holohedral  forms  are  geometrically  affected 
by  this  class  of  hemihedrism.  The  direct  and  inverse  forms 
are  superposable.  The  symbols  are  : 


r_w  z>**~i         rv**s  "P  ^?~i 
TT  {hk'li},  TT  {hklo},   and  ±  [_-J-J»  ±  I    ~^~ 

for  the  pyramids  and  prisms  respectively.  These  forms  are 
only  known  in  combination.  Fig.  151  is  an  example  in 

r,  £>3~\ 

an  Apatite  crystal,  containing  GO  P,  2  P,  P,  o  P,  2  P  2,    ^  —  I  . 

The  last  form  appears  only  on  the  right-hand  basal  solid 
angles  of  P,  or  in  the  corresponding  diagonal  zones  between 
P  and  oo  P. 

Tetartohedral  hexagonal  forms.  Supposing  a  dihexa- 
gonal pyramid  to  be  subjected  to  two  different  kinds  of 
hemihedrism,  the  resulting  form  will  have  only  one-fourth 
of  the  full  number  of  faces,  or  will  contain  two  out  of  the 
eight  groups  in  the  table  on  page  89. 

As  three  different  kinds  of  hemihedrism  are  possible, 
there  should  be  the  same  number  of  kinds  of  tetartohedrism, 
but  of  these  only  two  are  possible. 

i.  Trapezohedral  tetartohedrism  :  —  By  rhombohedral  he- 
mihedrism the  dihexagonal  pyramid  is  resolved  into  the  two 

scalenohedra  containing  the  groups  A—\  B  and  ^     3,  which, 

G  |  H          E  |  F 

by  a  further  plagihedral  development,  divide  into  the  four 

pairs   —  ,    -,    -,    -,  in  which,  when  the  faces  of  the  upper 
H     G    F     E 

group  are  even-,  those  of  the  lower  are  odd-numbered,  and 
vice  versa.  Geometrically  this  corresponds  to  taking  out 
the  faces  of  a  scalenohedron  by  pairs  adjacent  to  the  alter- 


CHAP.  IV.]        Trapezohedral  Tetartohedra. 


105 


nate  middle  edges,  as  in  fig.  153.  The  resulting  forms  (fig. 
152  from  the  white,  and  fig.  154  from  the  shaded  faces) 
are  called  trigonal  trapezohedra  ;  their  six  faces,  which  are 
trapezoids,  meet  in  six  polar  edges  all  of  the  same  length, 


FIG.  152. 


FIG.  153. 


FIG.  154. 


and  six  middle  ones  alternately  longer  and  shorter,  the 
longer  ones  being  extensions  of  edges  of  the  scalenohedron. 
They  have  the  same  axes  of  symmetry  as  the  scaleno- 
hedron— one  ternary  and  three  binary — but  no  planes  of 
symmetry,  and  are  therefore  not  superposable.  They  are 
distinguished  as  right-  and  left-handed,  positive  and  ne- 
gative, forms,  the  latter  having  reference  to  the  sign  of  the 
originating  scalenohedron.  Thus,  if  fig.  153  be  considered 
as  positive,  fig.  154  will  be  a  right-handed  and  fig.  152  a 
left-handed  positive  trapezohedron.  The  general  symbols 
are,  for  all  four  positions, 

,    mPn         ,    mPn  ,          mPn  mPn  , 

+  -     -  r,    +   -     -  /, —  r,    -  -     -/, 


or  KK'  \hkli\   if  K'  be  adopted  as  indicating  plagihedral 
hemihedrism.     The  first  and  third  and  second  and  fourth  of 


106  Systematic  Mineralogy.  [CHAP.  IV. 

these— that  is,  forms  of  similar  direction  and  opposite  signs 
— are  superposable. 

The  hexagonal  pyramid  of  the  second  order  may  be 
regarded  as  a  special  form  of  scalenohedron,  having  the 


FIG.  155. 


FIG.  156. 


\l 


/ 


angles  of  its  obtuse,  r,  polar  edges  =  180°,  and  its  middle 
edges  horizontal ;  and  therefore  if  half  its  faces  be  taken 
out  by  alternate  pairs  above  and  below,  having  a  middle 
edge  in  common,  and  the  remaining  ones  be  extended,  it 
will  satisfy  this  class  of  tetartohedrism.  The  form  is  a  tri- 
gonal pyramid  contained  by  six  isosceles  triangles  forming 
a  double  pyramid,  whose  base  is  an  equilateral  triangle. 
The  symmetry  in  regard  to  the  axes  is  therefore  the  same  as 
in  the  preceding  form :  the  lateral  axial  sections  as  well  as 
the  base  are  planes  of  symmetry.  The  derivation  of  the 
right-handed  form  is  shown  in  the  horizontal  projection 
(fig.  155),  and  that  of  the  left-handed  one  in  fig.  156;  but 
there  is  no  distinction  required  between  positive  and  nega- 
tive forms,  as  the  first  includes  both  the  positive  right  and 
negative  left  trapezohedra,  and  the  second  the  negative  right 
and  positive  left  ones.  The  symbols  are  : 

K  v>  }i  i  2  t}  and  (c  •/  {1 2  it},  or  -     -2  r  and  - 


CHAP.  IV.]         Trapezohedral  Tetartohedra.  107 

These  are  sometimes  written  as  -     -   or  \  m  P  2,  signifying 

that  they  contain  half  the  faces  of  the  originating  form, 
which,  though  convenient  as  indicating  the  character,  are 
otherwise  misleading,  as  they  are  not  hemihedral  forms ;  the 
pyramid  of  the  second  order  not  FlG  IS? 

being  susceptible  of  development 
into  a  solid  geometrically  dis- 
similar from  itself  by  any  of  the 
three  possible  methods  of  hemi- 
drism.  J'- 

The  dihexagonal  prism,  in  the 
same  way,  gives  rise  to  two 
tetartohedral  forms  called  ditri- 
gonal  prisms,  contained  by  three 
pairs  of  faces  making  alternately 
obtuse  and  acute  angles,  the  former  having  the  same 
values  as  those  meeting  the  lateral  axes  in  the  holohedral 
form,  as  is  seen  in  the  horizontal  projection  (fig.  157).  In 
combination,  they  bevel  the  alternate  edges  of  the  unit 
hexagonal  prism.  The  symbols  are  : 

<x>Pn     <x>Pn7 
K  K'  (h  klo},  or r,  —j-  l- 

The  prism  of  the  second  order  produces  two  exactly 
analogous  forms  known  as  trigonal  prisms,  contained  by 
three  vertical  faces  making  equal  angles  with  each  other, 
corresponding  in  position  to  figs.  155,  156.  The  symbols 
are : 

KK'(I  120),  or——  r,  ——-  /. 

Pyramids  and  prisms  of  the  first  order  do  not  give  any 
special  forms  by  trapezohedral  tetartohedrism. 

None  of  the  forms  of  this  class  ever  occur  independently 
or  otherwise  than  in  very  subordinate  combination,  and  they 
are  almost  exclusively  confined  to  one  species  ;  but  as  that 
is  the  most  abundant  of  all  minerals,  namely  quartz,  they 


io8  Systematic  Mineralogy.  [CHAP.  IV. 

are  of  considerable  interest,  fig.  157  a  being  a  characteristic 
example.     It  contains  R,  —  R,  <x>  P,  and  two  tetartohedra, 


2^2  r  (s) !  and  -— -  r  (x),  which  are    distinguished  as 

4  4 

right-handed  positive  forms  on  account  of  their  position 
to  the  right,  and  in  the  case  of  x,  below,  the  larger 
rhombohedron  R1}  considered  as  direct  or  positive.  In 

FIG.  157  a.  FIG.  157  *• 


fig.  157  b  the  same  faces  occur  to  the  left  of  R^  and  are 
therefore  left-handed  positive  forms.  The  trapezohedra 
are  negative  when  they  lie  below  the  faces  of  the  smaller  or 
negative  rhombohedron  r,  or  between  s  and/2  in  fig.  157  a,  or 
j  and  /6  in  fig.  157  l>,  the  right-  and  left-handed  character 
being  unchanged.  It  will  be  seen  from  the  figures  that  a 
face  s  lies  in  two  diagonal  zones  of  the  pyramid  or  with  the 
faces  R^pi  and/t  r2  in  fig.  157  a,  and  R^  /6,  and  r6/t  in 
fig.  157  b,  and  that  the  second  of  these  zones  in  either  case 
also  contains  a  face,  x,  of  a  trapezohedron.  No  single  crystal 
ever  contains  both  trigonal  pyramids,  or  right-  and  left- 
handed  trapezohedra  of  the  same  direction,  as,  although  a 
case  is  known  in  which  both  positive  and  negative  trapezo- 
hedra of  the  same  kind  form  an  apparent  scalenohedron,  the 
crystal  has  been  proved  by  optical  tests  to  be  a  compound 
or  twin  structure. 

1  These,  usually  known  as  the  rhomb  faces  in  quartz,  are  remarkable 
for  their  brilliancy,  whereby  they  may  often  be  detected  in  crystals  even 
of  microscopic  size. 


CHAP.  IV.]     Rhonibohedral  Tetartohedrism. 


Rhombohedral  tetartohedrism.  The  successive  applica- 
tion of  pyramidal  and  rhombohedral  hemihedrism  to  the 
dihexagonal  pyramid  corresponds  to  the  extension  of  alter- 


FIG.  158. 


FIG.  159. 


nate  faces  above  and  below  in  the  scalenohedron,  or  the 


groups 

' 


and 


?  divide  into  -,-  -  -,  each  of  which 

F  G    H'  E'  F 


G  I  H  E 

contains  six  faces,  either  all  even-  or  all  odd-numbered.  The 
resulting  form  is  a  rhombohedron,  whose  edges  do  not  lie 
in  any  of  the  principal  crystallographic  sections,  but  are 
oblique  to  the  lateral  axes,  as  shown  in  the  horizontal  pro- 
jections, figs.  158,  159.  This  is  known  as  a  rhombohedron 
of  intermediate  position,  or  of  the  third  order,  its  symbol 
being  K  -rr  \h  k  It] ,  or  the  whole  series,  according  to  Naumann, 

,    m  Pn  r          mPn  I 

~   j  >    i  —  ) 

4      /  A     r 


,  and  - 
4      /'  4 


in  which  positive  signs  represent  the  forms  derived  from  the 
positive  scalenohedron,  minus  signs  those  from  the  negative 
one,  and  the  letters  r  I  indicate  whether  the  right-  or  the 
left-hand  faces  are  above  or  below  the  middle  edges.  These, 
like  the  ordinary  rhombohedra,  are  all  superposable. 

The  hexagonal  pyramid  of  the  second  order  in  the  same 
way  gives  rise  to  two  rhombohedra,  whose  polar  edges 
lie  in  the  lateral  axial  planes,  or  make  angles  of  30  degrees 
with  those  of  the  ordinary  rhombohedron.  as  shown  in  figs. 


TIO  Systematic  Mineralogy.  [CHAP.  IV. 

1 60,  1 6 1.     These  are  said  to  be  of  the  second  order.    From 
the  analogy  of  the  preceding,  their  symbols  are  : 

mPz  r,    and   — — 2-. 

4     /  4     r 

The  dihexagonal  prism  gives  rise  to  two  hexagonal 
prisms  of  the  third  order,  oblique  to  the  axes,  whose  hori- 

FIG.  160.  FlG-  Ifil- 

& -4  *> ^ 


zontal  projections  will  be  apparent  from  the  contour  of  the 
rhombohedron  of  the  same  kind  previously  given,  as  either 
one  of  these  will  include  the  two  rhombohedra  upon  the 
same  base  originating  from  the  same  scalenohedron.  These 
are  exactly  similar  to  the  same  classes  of  prisms  originated 
by  pyramidal  hemihedrism. 

The  hexagonal  pyramid  and  rhombohedron  of  the  first 
order,  and  both  kinds  of  hexagonal  prisms,  do  not  give  rise 
to  any  special  tetartohedral  forms  of  this  class. 

The  tetartohedral  rhombohedra  only  occur  in  combina- 
tion, and  not  very  frequently.  Fig. 
161  a  is  a  case  observed  in  an 
Ilmenite  crystal  R.vR.—zR. 
P 2).  The  faces  of  the  last 
form  appear  only  on  the  left-hand 
polar  edges  of  R. 

The  impossible  method  of 
tetartohedrism  is  that  resulting  from  plagihedral  and  pyra- 
midal hemihedrism,  as  the  first  gives  forms  with  even-num- 


FIG.  161  a. 


CHAP.  IV.]  Miller's  Notation.  1 1 1 

bered  faces  above  and  odd  below,  or  vice  versa,  and  the 
second  requires  them  to  be  of  the  same  kind  above  and 
below.  The  successive  application  of  the  two  methods  to 
the  dihexagonal  pyramid,  therefore,  leaves  only  three  faces 
on  the  same  side  of  the  basal  section,  which  is  not  a  sym- 
metrical crystallographic  form. 

Miller's  rhombohedral  notation.  In  this  method,  the  forms 
are  referred  to  three  axes  making  equal  angles  with  each 
other,  and  having  equal  parameters.  These  axes  are  parallel 
to  the  polar  edges  of  the  unit  rhombohedron  of  the  series, 
and  are  therefore,  as  a  rule,  oblique  to  one  another.  The 
unit  form  (i  i  1}  is_  the  basal  pinakoid,  and  contains  two 
faces  (111)  and  (i  i  i).  Their  normal  is  called  the  axis  of 
the  rhombohedron  or  morphological  axis,  and  corresponds 
to  the  principal  axis  of  the  hexagonal  notation.  The  unit 
direct  rhombohedron  includes  the  three  pairs  of  faces  (i  o  o), 
(o  i  o),  (o  o  i),  and  the  corresponding  inverse  form  those 
with  the  indices  (221),  (121),  and  (122),  which  together 
give  the  hexagonal  pyramid  as  a  combination. 

The  general  form  h  k  /,  is  a  direct  scalenohedron,  and  the 
inverse  one  with  which  it  combines  to  form  the  dihexagonal 
pyramid,  is  distinguished  as  efg,  the  two  being  related  in 
the  following  manner  : — 

e  •=  2  (h  +  k  +  1)  —  $h  —  —  h  +  2  k  +  2l 

f=2(k   +   k  +   [)-$k  =      2k   -        k   +    2  I 
£•=2  (/*   +   £+/)   —   3/   =      2  h   +    2k  —       I 

The  unit  prism  has  the  faces  211,  121,  112,  211, 
121,  112;  the  prism  of  the  second  order,  i  o  i,  i~i  o, 
011,  i  o  i,  ii  o,  011;  and  the  dihexagonal  prism  those 
of  the  two  forms  (h  k  o)  and  (efo). 

The  hemihedral  forms  are  : 

1.  Asymmetric  ahkl,   corresponding  to  the    trapezo- 
hedral  tetartohedral  forms ; 

2.  Inclined  Khkl.     This  is  the  case  not  recognised  as  a 
symmetrical  kind  of  tetartohedrism  in  the  hexagonal  nota- 


112 


Systematic  Mineralogy.  [CHAP.  v. 

tion,  the  faces  of  the  form  being  all  either  positive  or  negative 
with  respect  to  the  principal  axis,  or  the  particular  class  of 
development  subsequently  noticed  as  hemimorphism. 

3.  Parallel  *  hkl,  corresponding  to  rhombohedral  tetar 
tohedrism  in  the  hexagonal  system. 

For  practical  purposes,  in  the  calculation  and  determina- 
tion of  crystals  this  method  is  generally  preferable  to  the 
hexagonal  notation,  as  it  dispenses  with  a  fourth  index  in 
the  symbols  ;  but  for  general  descriptive  purposes  it  does  not 
so  well  express  the  analogy  between  the  hexagonal  and 
tetragonal  systems,  and  it  has  therefore  not  been  adopted  in 
this  work.  The  student  will,  however,  do  well  to  become 
acquainted  with  Miller's  notation,  as  it  may  probably  super- 
sede the  hexagonal  form  at  no  very  distant  date.  A  simple 
exposition  of  it  will  be  found  in  Gurney's  elementary 
treatise  on  Crystallography. 


CHAPTER  V. 

TETRAGONAL1    SYSTEM. 

THE  complete  symmetry  of  this  system  is  contained  in  an 
upright  prism  upon  a  square  base,  which  has  quaternary 
symmetry  about  a  principal  axis  parallel  to  the  vertical 
edges,  and  binary  about  four  lateral  axes,  respectively 
parallel  to  the  sides  and  diagonals  of  the  base.  These  cor- 
respond to  five  planes  of  symmetry — a  basal,  or  principal 
plane,  and  four  lateral  planes  at  right  angles  to  the  first  and 
at  45°  to  each  other.  The  reference  axes  are  three,  at  right 
angles  to  each  other — namely,  the  vertical  or  principal  axis, 
c,  and  two  of  the  four  lateral  axes,  a},  az — those  parallel  to 
the  diagonals  of  the  base,  their  order  being  similar  to  that  of 

1  Other  names  are  Pyramidal,  Dimetric,  Quaternary,  Quadratic, 
and  Viergliedig  or  four-membered. 


CHAP.  V.] 


Ditetragonal  Pyramid. 


the  cubic  system.  The  parameters  of  the  lateral  axes  are 
similar,  and  different  from  that  of  the  vertical  axis,  the  two 
being  related  in  the  proportion  of  some  arbitrary  ratio  a  :  c, 
proper  to  the  species,  which  has  therefore  the  same  signifi- 
cation as  in  the  hexagonal  system. 

The  general  symbol  of  a  face  having  different  intercepts 
upon  the  three  reference  axes,  corresponding  to  different 
inclinations  upon  three  planes  of  symmetry,  none  of  which 
is  a  right  angle,  is  as  in  the  cubic  system  (h  k  /) ;  with  the 
difference  that  only  the  first  two  indices  are  interchangeable, 
giving  two  permutations  of  letters,  hk>  kh,  and  four  of  signs, 

+  +,  -\ — ,  — |-, ,  for  each  value  (positive  or  negative) 

of  /,  or  sixteen  in  all  as  the  maximum  number  of  faces 
possible  in  a  simple  tetragonal  form.  This,  known  as  a 
ditetragonal  pyramid,  represented  in  elevation  with  its 


FIG.  162. 


FIG.  163. 


symmetrical  sections  shaded  in  the  manner  described  on 
page  77,  in  fig.  162,  and  in  plan  in  fig.  163,'  is  a  double 
pyramid,  contained  by  sixteen  faces  whose  dihedral  angles 
in  the  eight  basal  edges  are  all  similar,  while  those  in  the 
polar  edges,  and  the  corresponding  plane  angles  of  the  equi- 


1  This  is  noted  as  P\. 
I 


Systematic  Mineralogy.  [CHAP.  v. 

lateral  eight-sided  base  are  alternately  larger  and  smaller. 
The  general  symbols  are  : 

{hkl},  (a  :  na  :  mc\  and  mPn. 

The  pyramid,  whose  base  is  a  regular  octagon,  and  has  the 
angles  of  its  polar  edges  all  equal,  is  an  impossible  form,  as 
requiring  for  n  the  irrational  value  tan.  67^°,  or  2-4142,  but 
FIG.  164.  for  any  rational  value  lower  than  this, 

the  more  obtuse  polar  edges  lie  in 
the  interaxial  planes,  and  when  n  =  i 
or  k  —  h,  their  angle  becomes  180°, 
or  the  two  faces  meeting  in  these 
planes  coincide.  This  corresponds 
to  the  tetragonal  or  square-based 
pyramid  of  the  first  order,  or  normal 
position  (figs.  164,  165) — 

\fih  1} ,  (a  :  a  :  m  c),  and  m  P, 

having  eight  faces  meeting  at  equal 
angles  in  the  four  basal  edges  and 
in  the  eight  polar  edges  which  lie  in 
the  lateral  axial  planes,  at  some  other 
angle  whose  difference  from  the  first  depends  upon  the 
disparity  in  length  between  the  vertical  and  the  lateral  axes. 
The  basal  angle,  ft,  corresponds  to  twice  the  plane  angle 
between  the  vertical  axis  and  a  lateral  interaxis  a',  and 
as  the  length  of  the  latter  is  to  that  of  the  adjacent  lateral 

axis  a,  inclined  to  it  at  45°,  as  i  :  — ^    the   fundamental 

ratio  a  :  c  for  any  pyramid,  assumed  as  the  unit  of  the  series, 
may  be  determined  by  the  expression — 

3      c  '2 

tan.  -  =  -.  and  c  =  — —--  when  a—\. 
2      a'  v  2 

when  the  measured  angle  is  that  over  a  polar  edge  =  TT,  the 


CHAP.  V.] 


Tetragonal  Pyramid. 


fundamental  parameter  of  c  is  found  by  computing  the  side 
/  opposite  to  that  angle  in  a  right-angled  spherical  triangle 
described  about  the  pole  of  the  principal  axis,  when — 

co tan.  -  =  cos./,  and  cotan./  =  c  when  a  =  i. 

2 

When  the  value  of  n  in  the  symbol  of  the  ditetragonal 


FIG.  165. 


FIG.  166. 


FIG.  167. 


pyramid  exceeds  tan.  67^°,  the  more  obtuse  polar  edges  lie 

in  the  lateral  axiaJ  planes,  and  when  n  =  <x>  or  k  =  o,  the 

angle  becomes  180°,  or  the  adjacent 

faces  meeting  in  them  fall  into  the 

same   plane,  giving   the    tetragonal 

pyramid    of  the    second    order   or 

diagonal   position   (figs.    166,    167), 

whose  basal   edges   are   equal   and  --& 

parallel    to    the    lateral     crystallo- 

graphic  axes,  while  its  polar  edges 

lie   in   the  lateral  interaxial  planes. 

The   symbols,  as  will  be  apparent 

from  its  derivation,  are — 

{hoi},  (a  :  oo a  :  mc\  and 
The  relations  of  these  three  classes  of  pyramids  are  similar 

I   2 


Systematic  Mineralogy. 


[CHAP.  V. 


FIG.  i 


to  those  subsisting  between  the  allied  forms  in  the  hexagonal 
system,  the  difference  being  in  the  possible  value  of  «,  which 
ranges  from  i  to  <x>  instead  of  merely  from  i  to  2. 

From  any  pyramid  of  either  kind  upon  the  same  base,  by 
multiplying  c  by  any  rational  quantity  m,  greater  or  less 
than   unity,  a  series  of  new  pyramids  of 
varying  altitude  is  obtained,  as  in  fig.  168, 
which  may  be  noted  as 

P,  2/>  3/>{iii},  {221},  {3  31},  or 
.  {113},  {223},  {in},  or 
,  {112}  {in}  {332}, 
according  as  one  or  other  of  the  three  is 
adopted  as  the  unit  form.  The  basal  angle 
increases  with  the  altitude  in  these  forms, 
and  when  m  =  co  or  /  =  o  it  becomes 
1 80°,  or  they  change  to  vertical  prisms 
of  unlimited  height.  There  are  therefore 
three  prisms,  one  corresponding  to  each 
kind  of  pyramid,  namely  — 

1.  Ditetragonal  prism,   {/z/£o},  (a  :  na  :  <x>c),  or  <x>Pn 
<fig.  169). 

2.  Tetragonal  prism  of  the  first  order,   \Jih  o}  =  \\.  i  oj, 
(a  :  a  :  GO  c),  or  oo  P  (fig.  170). 


FIG.  169. 

/^             001                  7| 

320 

—  ^ 

320 

230 

230 


3.  Tetragonal  prism  of  the  second  order,  {/zoo}  =  {100}, 
[a  :  GO  a  :  GO  c] ,  or  oo  Poo  (fig.  171). 

These  all  lie  in  a  zone  whose  plane  is  the  basal  section, 


CHAP.  V.] 


Tetragonal  Prisms. 


117 


FIG.  171. 


and  can  only  appear  in  combination.  In  the  first  the  angles 
of  adjacent  faces  are  alternately  larger  and  smaller,  and 
therefore,  from  their  measure,  the  characteristic  value  of  n 
may  be  deduced  ;  but  either  kind  of 
tetragonal  prism  has  all  its  angles 
right  angles,  and  cannot  be  used  for 
determining  the  parameters  of  the 
species,  except  when  in  combina- 
tion with  some  faces  not  meeting  it  MO  oi° 
at  right  angles.  In  the  other  direc- 
tion, when  m  is  less  than  i,  the 
basal  edges  become  sharper,  and 
the  polar  ones  more  obtuse,  as  it 
diminishes,  and  when  it  =  o  the  whole  of  the  faces  fall 
into  the  same  surface,  producing  an  unlimited  plane  called, 
as  in  the  preceding  system,  the  basal  or  terminal  pinakoid, 
and  having  the  symbols — 

{o  o  /}  =  {o  o  1} ,  (oo  a  :  oo  a  :  c]  and  o  P. 

This  can  occur  only  in  combination,  and  is  shown  as 
limiting  the  three  prisms  in  figs.  169-171  when  it  takes  the 
characteristic  shape  of  their  basal  sections. 

These  seven  kinds  include  all  the  possible  holohedral 
forms  :  their  relations  are  shown  in  Naumann's  system  by  a 
diagram  arranged  in  the  same  manner 
as  that  given  for  the  hexagonal  sys- 
tem, and  having  the  same  general 
signification.  The  only  difference 
between  it  and  the  table  on  page  84 

is  in  the  range  of  the  co-efficient  n      P Pn... 

from  i  to  oo,  while  in  the  former  it  is 
restricted  between  the  limits  i  and  2. 

In  combination,  forms  of  the  same 
order  will  appear  in  the  succession 
shown  in  the  vertical  lines  of  the  above  table,  the  steepest, 


P....oP. of 


-P...-Pn...L 


mP...mPn...  mP<x> 


Systematic  Mineralogy. 


[CHAP.  V. 


the  prism,  being  in  the  middle,  and  the  flattest,  the  terminal 
pinakoid,    at    the  ends  :  those  of  in- 

FIG.  172.  ....  .  . 

termediate  inclination,  the  pyramids 
proper,  being  arranged  in  regular  order 
towards  .either  of  these  limiting  forms, 
the  steeper  truncating  the  basal  edges 
of  the  flatter  ones,  and  conversely  the 
latter  truncating  the  polar  summits  of 
the  former,  as  in  fig.  172,  which  is  noted 
asoo/>,  f  P,  P, 


The  pyramid  of  the  second  order  truncates  the  polar  edges 
of  that  of  the  first  order,  when  both  are  of  the  same  altitude, 
or  have  m  in  common,  as  seen  in  elevation  and  plan  in 
figs.  173,  174,  which  also  illustrate  the  converse  case  of  the 
faces  of  the  pyramid  of  the  first  order  bevelling  the  solid 


FIG.  173. 


FIG.  174. 


angles  of  that  of  the  second  order.  When  the  two  forms 
differ  in  the  value  of  m,  if  m  Poo  is  steeper  it  will  bevel  the 
basal  solid  angles  of  P,  as  in  P,  2  P<x>,  fig.  175  ;  but  when 
m  P<x>  is  the  flatter  form,  it  will  truncate  the  polar  edges  ofP 
obliquely,  forming  four-faced  points  upon  the  polar  summits, 
as  in  P,  ^Pcc,  fig.  176.  A  ditetragonal  pyramid,  mPn, 
will  have  its  basal  solid  angles  in  the  lateral  interaxial 
planes  bevelled  by  the  faces  of  a  pyramid  of  the  first  order 
of  the  same  altitude,  or  will  bevel  the  polar  edges  of  the  latter, 
as  in  P,  P  3,  fig.  17  7 ;  and  similarly,  the  pyramid  of  the  second 


CHAP.  V.]  HoloJiedral  Tetragonal  Combinations.         119 


order  bevels  the  other  basal  solid  angles,  or  those  in  the 
lateral  axial  planes  of  mPn.  A  ditetragonal  pyramid  forms 
eight-faced  points  upon  the  principal  summits  of  a  steeper 
tetragonal  pyramid  of  either  order,  as  in  P%,  P$,  fig.  178  ; 


FIG.  175. 


FIG.  176. 


when  the  latter  is  the  obtuser  form  it  reduces  the  same 
summits  of  m  Pn  from  eight-  to  four-faced  ones,  as  in  3  P^, 
Pec,  fig.  179.  This  class  of  combination  is  of  considerable 
interest,  as  when  a  :  c  approximates  to  2  :  i,  the  solid  repre- 

FIG.  177.  FIG.  178. 


sen  ted  by  4^2  Por  {42  1}  {i  i  i},  is  almost  indistinguish- 
able from  the  trapezohedron  221}  of  the  cubic  system. 
This  case  actually  arises  in  Leudlte,  where  a  :  c  =  i  :  0-5264, 
whose  crystals,  until  recently  held  to  be  the  most  typical 
examples  of  the  particular  trapezohedron  in  question,  have 
been  shown  to  be  probably  assignable  to  a  tetragonal  com- 
bination of  the  character  of  fig.  179.  One  of  the  simplest 
cases  of  a  combination  of  tetragonal  prisms  and  pyramids  is 


I2O 


Systematic  Mineralogy. 


[CHAP.  V. 


shown  in  figs.  180,  181.     It  contains  oo  P,  GC/>OC,  P,  o  P, 
and  is  a  common  form  of  crystal  in  Apophyllite. 

As  in  the  hexagonal  system,  the  number  of  species  with 
full  tetragonal  symmetry  is  comparatively  small,  but  among 


FIG.  179. 


FIG.  180. 


FIG.  181. 


these  one,  namely  Idocrase,  is  remarkable  for  the  large 
number  of  forms  that  are  sometimes  combined  in  single 
crystals. 

Hemihedral  tetragonal  forms.     The  faces  of  the  dihexa- 
gonal  pyramid,  when  arranged  in  the  following  order  : 


A 

B 

C 

D 

i.  hkl 
v.  hkl 

n.  khl 
vi.  khl 

in.  khl 
vn.  khl 

IV.   hkl 

VIIL  hkl 

E 

F 

G 

H 

ix.  hkl 
xin.  hkl 

x.  khl 
xiv.  khl 

xi.  khl 
xv.  khl 

xn.  hkl 
xvi.  hkl 

— where  the  Roman  numeration  corresponds  to  that  in  fig. 
182,  may  be  symmetrically  halved  in  three  ways,  giving  rise 
to  three  kinds  of  hemihedral  forms  analogous  to  those  of  the 
hexagonal  system,  the  only  difference  being  that  each  group 
contains  two  instead  of  three  faces. 


CHAP.  V.] 


Tetragonal  Hemiliedrism. 


121 


Trapezohedral  hemihedrism.     The  first  method  of  selec- 
tion, that  by  alternate  groups  both  above  and  below,  or  the 


FIG.  182. 


(~*  T>      1       T\ 

and  -  — ,  the  former  containing  faces 


arrangements 

F  |  H  E  |  G 

which  are  all  odd-numbered  above,  and  all  even-numbered 


FIG.  183. 


FIG.  185. 


below,  while  in  the  latter  the  order  is  reversed,  gives  the 
forms  figs.  183,  184,  the  first  originating  from  the  white,  and 
the  second  from  the  shaded,  faces  of  fig.  184.  These,  known 


122  Systematic  Mineralogy.  [CHAP.  V. 

as  tetragonal  trapezohedra,  have  eight  faces,  meeting  four 
equal  polar  edges,  at  either  end  of  the  principal  axis,  and 
eight  unequal,  alternately  longer  and  shorter,  middle  ones  in 
zigzag  order  about  the  basal  section. 

From  these  figures  and  their  horizontal  projections  (figs. 
186,  187),  it  will  be  apparent — ist,  that  as  none  of  these 


FIG.  i 


edges  lie  in  planes  of  symmetry,  the  forms  are  plagihedral  ; 
and  2nd,  that  as  the  principal  solid  angles  are  formed  by  the 
meeting  of  four  similar  edges,  and  the  lateral  axes  and  inter- 
axes  bisect  the  middle  edges,  the  number  and  kinds  of 
axes  of  symmetry  are  the  same  as  in  the  holohedral  form, 
namely,  one  principal  or  quaternary,  and  four  binary. 
The  symbols  are  : 

a  {hkl}  {£/*/},  and  m-*^r 


No  other  holohedral  form  than  the  ditetragonal  pyramid  gives 
hemihedra  of  this  class  distinguishable  from  itself,  and  it 
is  not  known  whether  this  particular  kind  of  hemihedrism 
actually  occurs  or  not.  Its  existence  has  been  inferred  upon 
physical  grounds  in  a  few  salts  of  organic  bases,  the  most 
pronounced  example  being  a  sulphate  of  strychnine,  but  no- 
actual  plagihedra  of  the  above  kind  have  as  yet  been  ob- 
served with  certainty. 

SpJienoidal  hemihedrism.     The  second  method  of  selec- 
tion, that  by  alternate  pairs  of  faces  both  above  and  below, 

gives  the  arrangements  A     B  and  --LEL  the  first  contain- 
G     H  E     F 


CHAP.  V.] 


Tetragonal  Scalenohedra. 


123 


ing  odd-numbered  (ist  and  3rd)  pairs  above,  and  even-num- 
bered ones  (6th  and  8th)  below  ;  and  the  second  the  2nd 
and  4th  pairs  above,  and  the  5th  and  7th  below,  correspond- 
ing to  the  rhombohedral  selection  of  the  hexagonal  system. 
When  applied  to  the  ditetragonal  pyramid  it  produces  the 
forms  figs.  1 88,  190,  the  former  from  the  white,  and  the  latter 
from  the  shaded,  faces  of  fig.  189.  These,  known  as  tetra- 
gonal scalenohedra,  are  contained  by  eight  faces  meeting 


FIG. 


FIG.  18 


FIG.  190. 


in  three  prominently  dissimilar  kinds  of  edges.  Each  of 
the  principal  solid  angles  is  formed  by  two  longer  and  two 
shorter  polar  edges  ;  the  former  are  those  of  the  holohedral 
form,  and  have  the  same  dihedral  angles ;  the  third  kind, 
or  middle  edges,  those  of  the  zigzag  middle  belt,  represent 
alternate  middle  edges  of  the  tetragonal  trapezohedron,  and 
are  bisected  by  the  lateral  crystallographic  axes.  On  the 
same  side  of  the  base  the  longer  and  shorter  polar  edges  lie 
alternately  in  planes  at  right  angles  to  each  other ;  but  on 
opposite  sides  they  are  in  the  same  planes,  that  is,  in  the 
lateral  interaxial  sections,  which  are  therefore  the  only 
planes  of  symmetry.  The  axial  symmetry  is  binary  about 
the  three  crystallographic  axes. 


124  Systematic  Mineralogy. 

The  symbols  are  : 


[CHAP.  V. 


K  {hkl} 


,  and  + 


mPn         mP 


n  : 


the  direct  and  inverse  form  of  the  same  origin  being  super- 
posable  by  rotation  through  90°  about  the  principal  axis. 

The  tetragonal  pyramid  of  the  first  order  in  the  same 
way  gives  rise  to  figs.  191, 193,'  the  first  from  the  white,  and 
the  second  from  the  shaded,  faces  of  fig.  192.  These,  known 


FIG.  191. 


FIG.  192. 


FIG.  193. 


as  tetragonal  sphenoids,  from  their  wedge-like  appearance, 
are  obviously  only  special  cases  of  tetragonal  scaienohedra, 
having  the  obtuse  polar  angles  =  180°  ;  but  as  they  are 
more  frequently  met  with  than  the  latter  forms  they  are  con- 
sidered as  most  characteristic  of  this  kind  of  hemihedrism, 
which  is  therefore  called  sphenoidal.  The  shorter  or  hori- 
zontal edges  represent  the  shorter  polar  edges  of  the  sca- 
lenohedron,  and  the  longer  ones,  which  are  parallel  to  the 


1  These,  as  well  as  figs.  188-190,  are  on  a  smaller  scale  than  the 
holohedral  form. 


CHAP.  V.] 


Tetragonal  Sphenoids. 


12$ 


axes  of  the  diagonal  zone  in  the  pyramid,  the  middle  edges 
in  the  same  form.     The  symbols  are  : 

K  {khl} 


Like  the  scalenohedra,  they  are  superposable,  and  have  the 
same  planes  and  axes  of  symmetry. 

The  remaining  holohedral  forms  are  not  changed  by  this 
hemihedrism. 

The  wedge-like  character  is  not  apparent  in  the  forms 
generally  observed,  as  they  mostly  originate  from  pyramids 
which  are  either  very  obtuse  or  approximate  to  a  regular 
octahedron,  in  which  latter  case  the  sphenoid  is  very 
similar  to  a  regular  tetrahedron.  This  is  seen  in  copper 
pyrites,  which  is  the  only  characteristic  example  of  this 
hemihedrism  among  minerals,  the  combination  of  the  two 

P      P 

sphenoids  -  —  -  (fig.  194),  being  very  like  a  slightly  dis- 
torted regular  octahedron.  This,  however,  is  only  one  out 
of  many  sphenoids  found  in  the  same  mineral.  The  flatter 
ones  have  the  more  characteristic  combinations  shown  in  fig. 
195,  where  the  faces  of  one  sphenoid  truncate  the  edges 
between  oo  P  and  o  P  alternately  above  and  below.  Neither 


FIG.  194. 


FIG.  195. 


100 


sphenoids  nor  tetragonal  scalenohedra  ever  occur  indepen- 
dently, and  the  latter  when  present  truncate  the  solid  angles 


126  Systematic  Mineralogy,  [CHAP.  V. 

between  the  prism  and  sphenoid  faces  obliquely,  but  the 
faces  are  usually  very  small. 

Pyramidal  hemihedrism.     The  third  method  of  selection, 
that  by  alternate  pairs  of  faces  adjacent  to  the  same  basal 

edge,  gives  the  arrangements   — j—  and  -       ,  the  first  con- 

E  |  G          F  |  H 

taining  all  even-numbered  faces,  and  the  second  all  odd- 
ones,  corresponding  to  two  tetragonal  pyramids,  which  are 


similar  in  form  to  the  holohedral  ones,  but  not  in  position  ; 
the  sections  upon  planes  passing  through  the  polar  edges 
lying  in  one  case  to  the  right,  and  the  other  to  the  left  of 
the  lateral  axial  and  interaxial  planes  of  symmetry,  as  seen 
in  the  horizontal  projections,  figs.  196, 197,  which  also  show 
that  the  basal  edges  are  half  those  of  the  ditetragonal 
pyramid ;  the  symmetry  is,  therefore,  to  a  single  plane  the 
base,  and  quaternary  to  a  single  axis  the  principal  one. 

These  forms,  which  are  superposable,  are  known  as  tetra- 
gonal pyramids  of  the  third  order,  or  of  intermediate  posi- 
tion, the  symbols  being  : 

(7771  ( 7,  /    /"»  J       ,         f#2  P  M~\  Vm  P  H~\ 

TT  [h  kl\   TT  [k  h  1} ,  and  +      — 

L     2     J          L     2     J 

The  ditetragonal  prism  in  the  same  way  gives  two  prisms 
of  the  third  order,  whose  basal  sections  are  the  same  as  those 
of  the  corresponding  pyramid.  They  have  the  symbols  : 

IT  {A**}  *{***},  and  + 


CHAP.  V.]  Tetragonal  TetartoJiedra.  127 

v 

The  above  are  the  only  special  geometrical  forms  pro- 
duced by  this  hemihedrism  ;  they  FlG  Iq 
do  not  occur  independently,  but 
only  in   combinations,   and   are 
characteristic  of  a  small  but  well- 
defined   group   of   minerals,  the 
Tungstates  and  Molybdates.  One 
of  the  simplest  examples  is  given 

in  fig.  198,  a  crystal  of  molybdate  of  lead,  in  which  a  tetra- 
gonal prism  of  the  third  order  ^{430}  or  I  ^ — 5  J  trun- 
cates the  basal  edges  of  the  unit  pyramid  obliquely  to  the 
right  hand  of  the  lateral  axes. 

Tetartohedral  tetragonal  forms.  The  faces  of  the  ditetra- 
gonal  pyramid  may,  by  the  successive  application  of  two 
kinds  of  hemihedral  selection,  be  divided  into  four  symme- 
trical groups,  giving,  as  in  the  hexagonal  system,  two  possible 
classes  of  tetartohedra,  corresponding  to  the  following 
divisions : — 

Hemihedrism    .  .     i.     11.    in.  iv. 

Plagihedral  and  sphenoidal    -,     -,    -,     - 

H        G      E      F 

Sphenoidal  and  pyramidal .     -,    -  ,    -,    - 

G       H        F       E 

The  faces  to  which  these  correspond  will  be  seen  in  the  table 
on  page  120. 

The  forms  corresponding  to  the  first  of  these  divisions 
are  sphenoids,  differing  from  those  derived  by  hemihedrism 
from  the  tetragonal  pyramid  of  the  first  order,  in  the  position 
of  their  horizontal  polar  edges,  which  do  not  lie  in  planes  of 
symmetry,  but  cross  each  other  obliquely,  so  that  the  faces 
are  scalene  instead  of  isosceles  triangles,  and  are  not  sym- 
metrical to  any  principal  section,  while  preserving  the  same 
axes  of  symmetry  as  the  sphenoid,  the  relations  in  the  latter 
respect  being  similaj:  to  those  between  the  hexagonal  rhom- 


128  Systematic  Mineralogy.  [CHAP.  VI. 

bohedra  and  plagihedral  tetartohedra.  The  pyramid  of  the 
second  order  in  the  same  way  becomes  a  horizontal  prism, 
whose  section  is  the  rhomb,  having  for  its  diagonals  the 
vertical  and  a  lateral  axis.  The  dketragonal  prism  gives 
others  of  rhombic  sections,  whose  diagonals  are  in  the  propor- 
tion of  a  :  n  a,  and  the  prism  of  the  second  order  gives  two 
parallel  pairs  of  faces.  By  the  second  method  the  ditetra- 
gonal  pyramid  produces  sphenoids  of  the  same  geometrical 
properties  as  the  hemihedral  ones,  but  differently  placed  with 
respect  to  the  axes,  and  which,  like  the  analogous  tetarto- 
hedral  rhombohedra,  may  be  said  to  be  of  the  third  order. 
The  pyramid  of  the  second  order  gives  sphenoids  of  the 
second  order,  having  their  horizontal  edges  parallel  to  the 
lateral  axes  ;  and  the  ditetragonal  prism  gives  tetragonal 
prisms  of  the  third  order.  None  of  these  tetartohedral 
forms  have  as  yet  been  found  either  in  natural  or  artificial 
crystals  ;  but  they  are  interesting  as  geometrical  possibilities, 
and  as  showing  the  complete  analogy  subsisting  between  the 
hexagonal  and  tetragonal  systems  in  all  their  modifications. 


CHAPTER    VI. 

RHOMBIC  '    SYSTEM. 

THE  forms  of  this  system  are  referred  to  three  rectangular 
axes,  whose  parameters  are  all  different.  The  three  princi- 
pal sections  are  planes  of  symmetry  as  in  the  cubic  and 
tetragonal  systems,  but,  on  account  of  the  dissimilarity  of 
the  parameters,  the  symmetry  about  the  axes  is  only 
binary.  These  properties  are  apparent  in  a  vertical  prism 
of  definite  height  upon  an  oblong  rectangular  base,  whose 
length,  breadth,  and  depth  are  all  different,  which  is  only 
symmetrical  to  its  faces  and  about  its  edges,  and  the  lengths 
of  the  latter  are  proportional  to  the  parameters.  In  the 

1  Other  names   are,    Orthorhombic,    Orthoclinic,   Prismatic,   Tri- 
metric,  Terbinary,  and  Zweigliedrig. 


CHAP.  VI.] 


Rhombic  Pyramid. 


129 


general  symbol  h  k  I  the  order  of  the  indices  is  invariable, 
and  the  different  faces  are  represented  by  sign  permutations 
only,  giving  eight  as  the  largest  number  of  faces  that  can 
appear  in  any  simple  form.  The  forms  represented  by  hkk 
and  h  h  h  are  also  of  the  same  geometrical  character,  and 
as  the  crystallographic  elements  are  most  readily  deduced 
from  these  more  special  kinds,  it  is  convenient  to  consider 
them  first  rather  than  the  general  form.  The  unit  of  any 
series  is  a  rhombic  pyramid  or  octahedron,  such  as  fig.  199, 
which,  if  represented  by  (i  i  1}  a  :  b  :  c  or  P,  has  the  para- 
meters of  OA  =  a  =  6,  O £  =  b=.io,  and  O  C  =  c  =  8. 


FIG.  200. 


The  principal  sections  are  all  rhombs  of  different  propor- 
tions, their  deviation  from  a  square  form  increasing  with  the 
disproportion  of  the  parameters.  It  is  customary  to  place 
the  form,  so  that  the  longitudinal  axis,  a,  is  the  shorter 
diagonal  of  the  basal  section  and  the  right  and  left  one,  b, 
the  longer,  and  to  call  the  first  the  brachy diagonal  and  the 
second  the  macrodiagonal  axis,  the  third  vertical  axis  being 
noted  by  c  as  in  the  preceding  systems.  To  express  the 
ratios  of  the  parameters,  one  of  them,  usually  that  of  the 
macrodiagonal,  is  put  =  i:  thus,  in  fig.  199,  a  :  b  :  c  = 
o-6  :  ro  :  o%8;  but  if,  as  some  authors  prefer,  the  brachy- 


130 


Systematic  Mineralogy. 


[CHAP.  VI. 


FIG.  201. 


diagonal  is  considered  as  the  unit  a:b:c=i  :  i  -666  :  i  -333. ' 
The   choice  of  position  in  a  rhombic  pyramid  is  entirely 

arbitrary,  as  there  are  no 
peculiarities,  either  morpho- 
logical or  physical,  making 
any  one  axis  a  principal 
one,  so  either  one  may  be' 
made  vertical  or  horizontal 
at  pleasure.  Thus,  turning 
fig.  199  about  a  until  b  is 
vertical  gives  fig.  200,  when 
a  :  b  :  c=6  :  8  :  10  =  075  :  i  :  1-25;  and  turning  it  about 
b  gives  the  position  of  fig.  201,  where  the  shortest  axis  is 
vertical  and  a  :  b  :  <r  =  o'8  :  i  :  o-6,  all  of  which  ratios  have 
the  same  value. 

The  choice  of  one  of  these  positions  as  the  normal  one 
may  be  influenced  by  considerations  based  upon  structural 
peculiarities,  such  as  cleavages,  analogies  derived  from  similar 
forms  proper  to  other  species  of  like  constitution,  or  other 
more  arbitrary  methods  ;  but  no  definite  rule  can  be  laid 
down,  and  indeed  the  same  crystal  may  be,  and  often  is, 
differently  described  by  different  observers.  Schrauf  has 
suggested  that  the  first  median  line  of  the  optic  axes  should 
be  taken  as  the  vertical  axis,  but  the  suggestion  can  only  be 
adopted  where  the  crystal  is  transparent  and  susceptible  of 
optical  examination.  The  edges  of  the  rhombic  pyramid 
are  of  three  kinds — namely,  four  basal,  four  longer,  and  four 
shorter  polar  ones.  As  the  planes  bisecting  the  dihedral 
angles  between  the  principal  sections  are  not  planes  of 
symmetry,  the  measurement  of  the  angle  in  the  basal  edges 
is  not  sufficient  to  determine  the  parameter  of  c,  one  of 
either  polar  kind  being  required  in  addition.  Calling  (in 
fig.  199)  z  the  angle  over  a  basal  edge,  Y  and  x  those  over  a 
longer  and  shorter  polar  edge  respectively,  and  the  plane 
angle  of  z  upon  a,  and  those  of  x  and  Y  upon  b,  ft,  and  y, 

1  Naumann  adopts  the  first  order  and  Dana  the  second. 


CHAP.  VI.]        Holohedral  RJwmbic  Forms.  131 

the  relation  of  these  angles  to  the  parameters  are  expressed 
by  the  following  formulse : 

-,.  COS.  iy  X  /•>          COS    ^  Z 

Given  x  and  Y    cos.  a  =  -^ —  cos.  fi  =    . — ^- 

sm.  $  z  sin.  TT  x 

COS.  ^  Y  COS.  i  Z 

Y  and  z    sm.  a  =  ^ —  cos.  y  =  — . — f- 

sin.  ^  z  sin.  f  Y 

,  •       n          COS.  i  Y  •  COS.  ^  X 

x  and  Y     sin.  p  =    .     n2         sm.  y  =     .     *— 
sm.  7,  x  sm.  -|  Y 

— and  when  b  =  i,a  =  cotan.  «  =  c  tan.  y,  and  c  =  tan.  y  = 
a  tan.  /3.  In  the  case  of  prismatic  forms  the  angles  a,  /3,  and 
y  are  found  by  direct  measurement,  and  as  forms  of  this 
kind  are,  as  a  rule,  of  very  frequent  occurrence,  it  is  cus- 
tomary to  give  the  obtuse  angle  of  the  prism  as  a  chief 
characteristic  in  describing  rhombic  mineral  species. 

From  any  unit  pyramid,  by  changing  the  value  of  c  as 
before,  new  ones  are  obtained  of  varying  altitude  upon  the 
same  base,  such  as  cz,  c-A  (fig.  202),  ranging  upwards  into  the 
prism  and  downwards  into  the  basal  pinakoid  as  in  the  pre- 
ceding systems.  These  constitute  the  principal  or  vertical 
series,  and  are  distinguished  by  the  symbols — 

{oo.}       {hhl}  //</    {111}    {hhl}  h>l    {no} 

oo  a  :  GO  b  :  c     a  :  b  :  —  c      a  :  b  :  c    a  :  b  :  me      a  :  b  :  <x>c 
m 

oP          —P  P        m  P  ccP 

Any  pyramid  of  a  principal  series  will,  by  successively 
lengthening  its  brachydiagonal  axis  by  any  rational  multiples 
n.  produce  new  pyramids  all  of  the  same  height  and  length 
on  the  axes  a  and  c,  but  increasing  in  breadth  on  b  as  n 
becomes  greater,  as  seen  for  2b  and  3$  in  fig.  203;  and  when 
n  =  QO  the  form  is  a  rhombic  prism,  whose  edges  are  hori- 
zontal and  parallel  to  the  axis  b,  the  axes  a  and  c  being  the 
diagonals  of  its  base.  Prisms  of  this  kind  are  called  domes 
or  domas,  from  their  resemblance  to  house  roofs,  and  the 

K2 


132 


Systematic  Mineralogy. 


[CHAP.  VI. 


particular  one  in  question  is  known  as  the  macrodome,  its 
zone  axis  being  the  macrodiagonal.     If  this  is  derived  from 


FlG.   202. 


the  unit  pyramid,  it  will  be  the  unit  macro- 
dome;  but  as  the  same  development  applies 
equally  to  any  of  the  pyramids  of  the  prin- 
cipal series,  they  will  give  a  corresponding 
succession  of  macropyramids  and  limiting 
macrodomes,  the  whole  constituting  the 
transverse  prismatic  or  macrodiagonal  series 
— represented  by  Naumann's  general  symbols  mPn,  m  Pec, 
where  the  long  sign  signifies  that  n  applies  to  the  longer 
lateral  axis.  This  is  more  directly  indicated  in  Breit- 
haupt's  modification  of  the  symbol,  m  Pn,  which  is  therefore 
preferred  by  some  writers,  although  the  first  form  is  most 
generally  used.  The  order  of  symbols  for  the  principal 
types  of  this  series  is  as  follows  : — 

{hkl}  h>kh<l     {hkh}  h>k       {hkl}  h>k 

a  :  n  b  :  —  c  a  :  nb  :  c  a  \  nb  \  me 

m 


m 
[hoi] 


i 
m 


Pn  m  Pn 

{hoh}  {hoi} 

a  :  QO b  :  c     a  :  <x>fr  :  tt 


i 
m' 


mJ-'ac 


The  extension  of  the  brachydiagonal  of  any   unit  or 


CHAP.  VI.] 


Brachy diagonal  Series. 


133 


other  pyramid  of  the  principal  series — as  in  fig.  204,  where 
that  axis  is  successively  made  2  a,  3  a,  GO  a,  while  b  and  c 
are  unchanged — produces  a  third  class  of  pyramids  and 
domes  known  as  the  longitudinal  or  brachydiagonal  series, 
which  are  distinguished  in  their  symbols  from  the  transverse 
series  by  the  sign  ~,  indicating  the  shorter  lateral  axis,  placed 
over  the  characteristic  POT  its  coefficient,  thus  m  Pn  or  ;;/  Pn. 
The  order  of  the  typical  symbols  is  as  follows  : 

[hkl]  h<k<l         {hkk}  h<k          (hkl}h<kh>l 


n  a  :  b  :  —  c 
m 


n  a  :  b  :  c 


u  a  :  b  :  m  c 


—  Pn 

Pn 

mPn 

m 

{o/£/}  k<.l 

{okk} 

[okl]  k-. 

co  a  :  b  :  —  c 

aoa  '.  b  :  c 

ao  a  :  b  :  me 

m 

-^00 

Px> 

mPao 

m 

FIG.  204. 


The  prism  of  the  principal  series  having  its  lateral  axes 
in  the  unit  proportion  a  :  £,  will,  by  lengthening  either  of 


1 34  Systematic  Mineralogy.  [CHAP.  VI. 

these  axes  relatively  to  the  other,  give  rise  to  new  prisms, 
whose  characteristic  angle  increases  with  the  value  of  //. 
These  are  called  macroprisms,  when  derived  from  the  varia- 
tion of  b,  and  brachyprisms  from  a,  which  are  respectively 
represented  by  oo  Pn  and  <x>Pn.  When  n  in  either  series 
becomes  <x>,  the  angle  of  the  prism  is  180°,  or  it  is  reduced 
to  a  single  plane  parallel  to  one  lateral  axis  and  perpen- 
dicular to  the  other,  giving  the  two  forms  known  as  the 
macropinakoid  and  brachypinakoid ;  the  former  is  repre- 
sented by  (i  o  o}  a  :  oo  b  :  oo  c,  or  oo  Poo,  and  the  latter  by 
{010}  oo  a  :  b  :  oo  c,  or  oo  J*<x>. 

The  above  include  all  the  simple  forms  possible  in  the 
rhombic  system,  namely,  pyramids  of  eight  faces,  the  only 
closed  forms,  prisms  of  four  faces  in  three  positions,  and  the 
three  pinakoids  of  two  faces.  Their  relations  to  each  other 
may  be  shown  by  arranging  their  symbols  in  a  diagram  of 
the  kind  given  in  the  preceding  systems.  Here  the  unit 

oP oP. oP.....oP oP        form   P  in   the 

:  :  ;  centre  has    the 

i     7x          irt          i    r>      i  n          T«         obtuser  forms  of 
_  Pao...—Pn —  £. Pn. 


m  m 


A 


'm    :""»/:    '"'iu:  the  principal 

series  below  and 
.Pn...     .  P=c     the  acuter  ones 


:  above  it  in  the 

P<x>...mPn mP...mPn ;;•  P^o     same  vertical 

.  .  .  « 

•  -        line-     The  lines 
oc/>oo     next     to    right 

and  left  contain  the  series  of  the  macropyramids  and 
brachypyramids  respectively,  and  the  first  and  last  lines,  the 
brachydomes  and  macrodomes.  The  top  horizontal  line 
contains  the  symbol  of  the  basal  pinakoid  as  the  common 
limit  of  all  the  series ;  the  other  pinakoids  are  at  the  ends 
of  the  lower  horizontal  line,  the  intermediate  positions 
being  taken  by  the  different  prisms.  As  in  the  previous 
system,  the  forms  contained  in  any  line,  whether  horizontal 
or  vertical,  lie  in  the  same  zone. 


CHAP.  VI.] 


Rhombic  Notation. 


135 


N     N    N    N    N     N     N 

M                pie,,.,.,  p.,,,  (.,.,„.,..,«,,.. 

8   8   8  ^8   8  ^8   8 

8 

g              H»-+»N|n        ««  N    ro 

""V|              «ln-*Nr#n        n::i  N    ro 

8             ---.... 

8 

-  -^ 

O     rf    CO   N     ^    i-«     N 
N     N     P»     N    'O     M    \O 

O               ON  >O    CN  ro  N    ro  1-1 

O    O    O    o    O    O    O 

ro    £       •  —  '  •—  -  •  ^—  '  -^2-  •  —  "  —  ••  — 

§  3 

w  rt 
"—.S 

-+..X*   ""«««« 

8)17*     •^fcoH1^^^        P°I?I  r-i    ro 
HM 

OQ                                  «ln         ro 

8  § 

..  oj 

^^0-^.* 

^^      8   8   8   8   8,  8   8 

"*>  ^ 

8 

N     N     N     N     M     M     C4 

N                 ««««««« 

8                ««<3««<3« 

« 

ss^^: 

8 

Q^f^c^n^e^ix,^,!^, 

ro  ro  ro  ro  ro  ro  ro 
-i|«  -4-.]  inm        niu  N    ro 

ro 

O\  *O    O^  f)  W    ^O  HH 

o  e 

ro  .!£ 

roMroi-iN-.i-i     o 
«i-Ni-irONroi-i 

g        ONVOONror-troi-i 
'C        rorovOrorovOro 

0  s 

"     in 
ro  'S 

Hl.^*,  %X  «    r^ 

vT  J3 

8   ^ 

o     rt 

^^^^^•^•^ 

O 

,^       rororororororo 

ro 

ro  ro  ro  ro  ro  ro  ro 

ro 

«««««««  « 

«*«««.«« 

<3 

8   8 

S 

ft 

8 

S 

8 

8 

<,    IX, 

"M-Hw 

S 

?1?) 

s 

u. 

N 

>s 

ro 

8_ 

Hn-^nin 

«l«  N    ro 

8 

'5s,   '^  1^,   'R,   '-N   'ft,   l!^ 

8 

ro  rq 

o  o 

s? 

N 

0 

g 

r-l 
ro 

o 

<M 

0 

ro 

0 

-1 

'ON  ON  ON 
ro  ro  ^O 

"rol^  "ro"^ 
<^  rovo    ro 

"5"   p 

ro    in 

H    _,    M    1-1   ro  M    ro 

f  S 

"  — 

**-*  "_j 

•2,   N     TJ-    N 

N    'C 

M     *"" 

*r 

vT  c 

1    '    '~ 

—  .  —  .  — 

•~^       &! 

_    —               — 

—  a, 

r*:-: 

M 

<**. 

0         fl. 

^    -o"  C  C 

%T  F^1 

c  ^ 

••  ,0" 

HnHwwIM 

MIM  N    ro 

o      O 

^IcoH^w^n        ccjci  M    ro 

8    S 

-^   •* 

"* 

"^ 

"* 

"* 

* 

M 

^^^ 

^^ 

^« 

N    N    N    N    N    N    N 

^^ 

(3     « 

8   8 

8 

8 

8 

8 

8 

8 

«i^  KIM  cciw 

1,^^^ 

WM 

«   «  «  «  «  «  « 

^ 

If  all  the  columns  in  this  table  be  placed  side  by  side  in  the  same 
line,  it  will  form  substantially  an  extension  of  the  diagram  on  the 
preceding  page.  The  symbol  of  the  basal  pinakoid  common  to  all 
is  omitted. 


136  Systematic  Mineralogy.  [CHAP.  VI. 

From  the  small  number  effaces  possible  in  any  single  form, 
the  notation  of  rhombic  crystals,  especially  in  Naumann's 
method,  involves  a  considerable  diversity  in  the  symbols. 
In  illustration  of  this  point,  the  table  on  the  preceding  page 
gives  the  notation  of  a  face  of  mPn,  mPn,  mPn,  when 
m  is  successively  made  o,  ^,  -|,  §,  i,  £,  2,  3,  and  oo,  and  n  i,  f , 
2,  3,  and  oo,  from  which  it  will  be  seen  that  these  coefficients 
involve  the  use  as  indices  of  the  numbers  o,  i,  2,  3,  4, 6, 
and  9. 

In  combinations  of  rhombic  forms,  as  in  those  of  the 
preceding  systems,  any  pyramid  has  its  basal  edges  bevelled 
by  a  more  acute  one,  and  truncated  by  the  prism  ;  and  its 
polar  solid  angles  blunted  by  a  more  obtuse  pyramid  of  the 
same  series,  and  truncated  by  the  basal  pinakoid.  The 
polar  edges  in  the  brachydiagonal  section  are  bevelled  by 
any  macropyramid  and  truncated  by  a  macrodome,  having  a 
and  c  in  common,  and  the  other  polar  edges,  those  in  the 
macrodiagonal  section,  are  similarly  modified  by  the  brachy- 
pyramid  and  brachydome,  having  b  and  c  in  common  with 
the  pyramid.  A  prism  has  its  obtuse  F[G 

edges,  or  those  facing  the  macrodia- 
gonal, bevelled  by  brachyprisms,  and 
truncated  by  the  brachypinakoid  ;  and 
its  acute  edges  are  similarly  modified 
by  macroprisms  and  the  macropina- 
koid.  The  observed  combinations  are 
exceedingly  numerous  and  diversified 
in  appearance,  which  diversity  is  a 
consequence  of  there  being  only  lateral 
symmetry  to  the  axes  ;  and  although  the 
edges  in  all  the  pinakoid  sections  may  be  modified  by  faces 
of  the  same  kind,  it  is  more  general  to  find  them  differently 
modified  in  each  section.  For  instance,  in  the  three  figs.  205, 
206,  207,  the  first,  in  which  P  is  truncated  longitudinally  and 
vertically  by  the  pinakoids  oo  P<x,  and  oP,  and  transversely 
by  the  middle  edges  of  the  unit  brachydome  /oo,  and  the 


CHAP.  VI.] 


Rhombic  Combinations. 


137 


second,  where  a  joins  the  solid  angles  formed  by  the  crossing 
of  the  edges  of  the  unit  prism  GO  P  and  macrodome  Poo,  b, 
the  acute  edges  of  GO  P,  and  c,  the  faces  of  the  terminal  pina- 
koid — are  more  characteristic  cases  than  fig.  207,  where  all 


three  pinakoids  and  prismatic  forms  are  present.  These 
examples  are  taken  from  the  species  sulphur,  which  is  one 
of  the  few  minerals  in  this  system  whose  crystals  have  pyra- 
mids as  the  dominant  forms.  These  are  noted  as  P,  ^  P,  and 
A  P.  Fig.  205  is  the  common  form  obtained  when  sulphur 

FIG.  208.  FIG.  209. 


is  crystallised  by  the  spontaneous  evaporation  of  its  solution 
in  bisulphide  of  carbon. 

Figs.  207,  208  are  examples  of  combinations  without  the 
basal  pinakoid.     The  first,  drawn  to  the  elements  of  the 


138 


Systematic  Mineralogy.  [CHAP.  VI. 


species  Topaz,  contains  P,  <x>P,  00/2,  Poo,  oo/oo,  co /GO, 

most  of  these  forms  being  in  the  zone  of  the  principal  prism  ; 
while  fig.  209  is  chiefly  modified  in  the  longitudinal  prismatic 
or  brachydiagonal  zone,  which  contains  i/co,  /GO,  2/00, 


FIG.  210. 


FIG.  211. 


the  other  forms  being  P,  GO  P,  2  Pec,  GO  Pec  ;  this  is  drawn 
to  the  parameters  of  Brookite. 

Another  consequence  of  the  absence  of  diagonal  axes  of 
symmetry  is  the  tendency  to  elongation  parallel  to  one  axis, 
producing  solids  which  are  essentially  prisms,  a  peculiarity 


which  is  indicated  in  the  name  '  prismatic '  applied  by  Miller 
to  this  system;  and  as  the  extension  may  be  along  either  axis 
indifferently,  the  same  combination  in  the  same  substance 
may  appear  in  many  different  shapes,  according  as  one  or 
other  form  predominates.  Thus  in  fig.  210  the  three  unit 
prismatic  forms  (prism  and  domes)  are  combined  in  about 


CHAP.  VI.] 


Rhombic.  Combinations. 


equal  dimensions  with  the  three  pinakoids;  in  fig.  211  the 
solid  is  essentially  prismatic,  the  vertical  edges  being  the 
longest.  In  fig.  212  the  character  is  longitudinal,  prismatic 
or  brachydomatic,  the  greatest  length  being  parallel  to  a  ; 
and  in  fig.  213  it  is  transverse-prismatic,  or  macrodomatic, 
the  longest  edges  being  parallel  to  b.  These  variations 
are  quite  possible,  and  are  to  some  extent  represented  in 
the  species  Barytes. 


FIG.  214. 


FIG.  215. 


A  few  more  illustrations  of  the  simpler  class  of  combina- 
tions are  given  in  figs.  213-223,  from  the  closely  allied 
species,  Barytes  and  Anglesite,  which  are  remarkable  for  their 


FIG.  217. 


great  variety  of  forms,  these  being  selected  from  nearly  a 
hundred  described  crystals  of  these  minerals.  Fig.  214  is 
the  unit  macrodome  of  Barytes,  P  GO,  shortened  in  the  direc- 


140 


Systematic  Mituralogy. 


[CHAP.  VI. 


tion  of  its  axis  by  <x>  P  GO,  and  truncated  in  its  middle  edges 
by  oo  Pec.  Fig.  2 1 5  is  similar,  with  the  addition  of  ^  Pec  and 
o  P.  Fig.  216  is  Pac,  o  P,  with  the  middle  solid  angles  trun- 
cated obliquely  by  the  prism  oo  P,  and  the  upper  and  lower 


FIG.  218. 


FIG.  219. 


ones  by  the  brachydomes  Pec,  2  Poo,  4  P  oo  .  Fig.  217 
contains  2  ^oo,  4  Pec,  oo  ^oo,  elongated  parallel  to  a,  and 
limited  by  the  prism  <x>P.  In  fig.  218,  the  upper  edges  be- 
tween Pec  and  oo  Pec  are  truncated  obliquely  by  the  acute 
brachypyramid  3  P$,  and  those  between  ooPand  oo  ^oo 
in  front  by  a^more  acute  macrodome  2  Pec.  Fig.  219  is  the 


FIG.  220. 


FIG.  221. 


pyramid  2  Pof  Anglesite,  with  its  macrodiagonal  polar  edges 
truncated  by  2  Pec.     Fig.  220  is  a  more  acute  pyramid,  4  P 


CHAP.  VI.] 


Rhombic  Combinations. 


141 


with  the  same  brachydome,  and  the  unit  macrodome  modify- 
ing the  brachydiagonal  polar  edges.  Fig.  221  contains  oo  P, 
P<x>,  2pac,  co  Poo,  and  2  P,  the  latter  modifying  the 
solid  angles  formed  by  the  meeting  of  the  three  prismatic 
forms.  That  this  is  one  of  the  pyramids  of  the  principal 
series  is  apparent  from  the  horizontality  of  its  edges  of 
combination  with  the  prism. 

In  fig.  222  the  faces  of  the  macrodome  2  Pzc  are  so 


FIG.  222. 


FIG.  223. 


proportioned  as  to  form  rhombic  planes  truncating  the  front 
solid  angles  between  coP  and  P.  Fig.  223  is  of  the  same 
general  character,  but  the  basal  edges  of  the  prism  are 


FIG.  224. 


FIG.  225. 


modified  by  2  P ;  and  the  lateral  solid  angles  formed  by  P, 
GO  P,  and  GO  ^oo,  are  replaced  by  an  acute  brachypyramid 


142 


Systematic  Mineralogy. 


[CHAP.  VI. 


FIG.  226. 


2  p2,  whose  edges  of  combination  are  parallel  to  the 
macrodiagonal  polar  edges  of  the  pyramid.  Fig.  224  is  the 
unit  macrodome  Pec,  lengthened  parallel  to  its  axis,  limited 
laterally  by  2  Pec,  and  its  middle  edges  bevelled  by  |  Poo. 
Figs.  225,  226  are  examples  of  simple  combinations  of  Oli- 
vine.  The  first  is  ccp2,  GO  Pec,  2  Pec,  and  the  second 
<x>p2,  cc Pec,  P<x>,  the  latter  being  very 
commonly  observed  in  crystallised  slags 
obtained  in  puddling  and  heating  fur- 
naces. These  examples  will  suffice  to 
show  the  general  character  of  the  com- 
binations of  this  system,  but  they  are 
only  of  simpler  kinds.  For  those  of  more 
complex  character  the  reader  is  referred  to  the  larger  special 
memoirs  and  descriptions,  especially  to  Schraufs  atlas  of 
crystalline  forms. 

The  fundamental  parameters  of  any  rhombic  series  of 
crystals,  being  irrational  numbers,  they  may,  when  two  are 
nearly  equal,  produce  forms  approximating  in  character 
to  those  of  the  tetragonal  system,  but  the  true  nature  will 
usually  be  apparent  by  their  modi- 
fications. Fig.  221,  for  instance, 
would  be  nearly  like  the  common 
combination  Pec  P  in  Tinstone  but 
for  the  rhombic  faces  of  2  Pec. 

Where  the  parameters  of  the  axes 
a  and  b  are  related  to  each  other  in 
the  proportion  of  i  :  V  3.  the  obtuse 
angle  of  the  prism  will  be  120°,  and 
as  the  brachypinakoid  truncates  its 
acute  angles,  the  combination  of 
these  two  forms  will  be  an  equal 
six-sided  prism,  having  all  its  angles 
of  120°,  or  geometrically  identical 

with  the  unit  prism  of  the  hexagonal  system  ;  and  in  like 
manner  any  pyramid  of  the  same  series  combined  with  a 


FIG.  227. 


CHAP.  VI.] 


Rhombic  Hemihedrism. 


brachydome  of  twice  the  height — i.e.,  P  with  2  -Poc,  or  2  P 
with  4/^00 — will  produce  a  regular  hexagonal  pyramid. 
Fig.  221  is  an  example  of  such  a  combination  in  Witherite, 
which  is  very  similar  in  appearance  to  the  ordinary  form 
of  quartz  crystal.  Other  examples  are  afforded  by  Ara- 
gonite  and  Bisulphide  of  Copper,  where  the  prismatic  angle 
approaches  very  nearly  to  120°.  Such  forms  may,  however, 
as  a  rule,  be  discriminated  without  much  difficulty  by  the 
unequal  modification  of  their  edges,  peculiarities  of  cleavage, 
&c.,  and,  when  transparent,  by  their  optical  properties.  The 
solid  formed  by  the  combination  of  the  three  pinakoids  may 
also,  in  some  instances,  appear  very  like  a  cube ;  the  best 
example  of  this  is  afforded  by  Anhydrite,  or  anhydrous  Sul- 
phate of  Calcium. 

Hemihedral  rhombic  forms.  A  rhombic  pyramid  may, 
by  the  omission  of  alternate  faces  right  and  left,  above  and 
below  the  base,  give  rise  to  hemihedral  forms  analogous  to 
the  tetartohedral  sphenoids  of  the  tetragonal  system,  as 


FIG.  228. 


FIG.  229. 


FIG.  230. 


shown  in  figs.  228  and  230,  the  first  being  that  derived 
from  the  white  faces,  and  the  second  from  the  shaded  ones 
in  fig.  229.  These  differ  from  the  tetragonal  sphenoid  by 
the  inequality  of  the  middle  edges,  two  being  obtuse  and 
two  acute  ;  and  also  their  polar  edges  do  not  lie  at  right 
angles  to  each  other,  but  cross  obliquely,  the  angle  between 


144 


Systematic  Mineralogy. 


[CHAP.  VI. 


their  horizontal  projections  corresponding  to  the  acute  angle 
of  the  prism.  As  in  the  case  of  the  tetrahedron  and  tetra- 
gonal sphenoids,  they  are  not  symmetrical  to  rectangular 
axial  planes  ;  and  these  being  the  only  possible  planes  of 
symmetry  in  the  system,  they  are  plagihedral,  being  per- 
manently right-  and  left-handed,  according  to  their  origin. 
The  symbols  are : 

f  7     7     71  J        ,       m   P  J  W  P 

K  {hkl}  and  + and  —   . 

2  2 

The  prismatic  forms  are  not  affected  geometrically  by  this 
kind  of  hemihedrism,  which  is  not  of  very  frequent  occurrence 
in  natural  crystals.  The  principal  examples 
(fig.  231)  are  found  in  Sulphate  of  Magnesium 
and  the  isomorphous  Sulphate  of  Zinc. 

There  is  another  class  of  hemihedral  forms 
possible  in  the  rhombic  system,  but  only  a 
single  example  has  been  demonstrated  as 
existing  in  an  artificially  crystallised  com- 
pound. These  are  analogous  to  the  scaleno- 
hedral  or  rhombohedral  forms  of  the  pre- 
ceding systems,  being  produced  by  the 
alternate  development  of  pairs  of  faces 
preserving  one  of  the  original  edges.  The 
result  is  the  production  of  a  prism  upon  the  same  base 
as  the  pyramid,  whose  edges  are  inclined  instead  of  being 


FIG.  231. 


no 


FIG.  232. 


FIG.  233. 


vertical  or  horizontal,  the  inclination  being  similar  to  that 
of  the  edges  retained.     For  instance,  fig.  233  is  the  oblique 


CHAP.  VII.]  Oblique  Symmetry,  145 

rhombic  prism  of  this  kind  produced  from  the  faces  I.,  iv., 
vi.,  and  vii.,  in  fig.  232  ;  its  edges  are  parallel  to  the  more 
obtuse  polar  edges  in  the  pyramid,  and  its  faces  are  sym- 
metrical to  the  brachypinakoid  alone.  The  lower  front  and 
upper  back  pairs  of  faces  would  produce  a  similar  form, 
with  a  forward  slope,  but  symmetrical  to  the  same  plane, 
and  in  like  manner  from  the  extension  of  alternate  pairs  of 
basal  and  macrodiagonal  edges,  pairs  of  similar  prismatic 
forms  may  be  derived,  having  the  same  symmetry  to  one  axial 
plane  only.  From  this  latter  circumstance,  such  forms  are  said 
to  have  monosymmetric  hemihedrism,  a  property  which  they 
have  in  common  with  all  other  forms  of  the  next  system. 

Formerly  several  minerals  were  referred  to  this  type  of 
hemihedrism,  but  they  are  now,  upon  structural  considera- 
tions, placed  in  the  oblique  system. 


CHAPTER  VII. 

OBLIQUE1     SYSTEM. 

THE  forms  of  this  system  are  referred  to  three  axes  having 
dissimilar  parameters,  one  being  at  right  angles  to  the  other 

FIG.  234. 

c 


C 

two.     If  this  be   considered  as  the  axis  of  breadth,  and 
placed  horizontally,  as  B  B  in  fig.  234,  it  will  be  normal  to 

1  Other  names    are    clinorhombic,    monoclinic,    oblique-rhombic, 
binary,  monosymmetric,  and  zwei  und  eingliedrig. 


146 


Systematic  Mineralogy. 


[CHAP.  VII. 


an  upright  longitudinal   plane  containing   the   other   two 
axes,  which  may  be  oblique  to  each  other,  and  parallel  to 
two  others,  in  each  of  which  it  will  be  at  right  angles  to  one 
of  the  remaining  axes.    The  solid  whose  edges  are  shown  by 
the  fine  dotted  lines,  which  may  be  supposed  to  represent 
that  derived  from  the  unit 
parameters,  will  therefore 
be  rhombic  in  two  of  its 
principal     sections,     and 
rhomboidal  in   the  third, 
as   in    the    three    plane 
projections,  figs.  235,  236, 
237,  which  are   similarly 
noted    to    the  preceding 
one.     From  these  it  will 
be  seen  _that    the  rhom- 
boid AC  AC,   fig.    237,  di- 
vides the  solid  symmetrically  into   right  and  left  halves, 
whether  it  be  looked  at  from  the  front,  as  in  fig.  235,  or  from 
above,  as  in  fig.  236  ;  while  in  fig.  237  the  division  by  the 


other  two  planes  into  right  and  left  and  upper  and  lower 
halves  are  both  unsymmetrical.  This  property  of  symmetry 
to  a  single  longitudinal  plane  is  the  most  essential  character 
of  the  system,  and  there  is  usually  a  marked  obliquity  be- 
tween the  axes  in  that  plane,  which  as  a  necessary  crystallo- 


CHAP.  VII.]          Oblique  Hemipyramids.  147 

graphic  element,  in  addition  to  the  parameters  a,  b,  c,  is  ex- 
pressed as  the  angle  /3.  When,  as  is  generally  done,  one  of 
these  axes  is  placed  upright,  and  the  third  with  a  forward 
inclination,  as  in  fig.  234,  (3  is  considered  as  the  acute  angle 
in  front  below,  or  on  the  negative  side  of  the  vertical  axis  ; 
the  supplemental  obtuse  angle  (180°  —  /3)  being  on  the  posi- 
tive side  above,  which  positions  are  reversed  behind.  In 
this  order  the  axes  a  and  b  are  as  in  the  preceding  system 
diagonals  of  a  rhombic  section  ;  but  the  former  is  inclined, 
while  the  latter  is  horizontal  to  the  third  axis  c.  They  are 
therefore  distinguished  as  clinodiagonal  and  orthodiagonal 
axes.  In  the  arrangement  of  the  parameters  that  of  b  is  con- 
sidered as  unity,  but  it  is  not  necessarily  longer  than  that 
of  a. 

The  solid  under  consideration  is  geometrically  an  oblique 
rhombic  pyramid  or  octahedron,  but  it  is  not  a  simple  crys- 
tallographic  form,  being  made  up  of  two  dissimilar  classes 
of  faces  marked  by  longer  or  shorter  edges  in  the  plane  of 
symmetry,  according  as  they  face  the  obtuse  or  acute  angle 
of  the  inclined  axis,  and  either  set  may  occur  in  combination 
with  or  without  the  other.  It  is  therefore  to  be  considered 
as  contained  by  two  half  pyramids  or  hemipyramids,  one 
having  the  faces  n.,  in.,  v.,  and  VIIL,  as  in  fig.  234,  opposite 
the  acute  angle,  and  the  other  i.,  iv.,  vi.,  vn.,  facing  the 
obtuse  angle  of  the  axes.  For  the  general  forms  represented 
by  three  dissimilar  finite  indices,  the  two  groups  are 

hkl 


^ 
hkl\hkl 

where  the  stronger  letters  indicate  faces  in  front  of  the  ortho- 
diagonal  section.  Naumann  calls  these  positive  and  negative 
hemipyramids,  or  +  P,  and  —  P,  and  considers  the  former 
as  that  facing  the  acute  angle  of  the  axes,  which  conven- 
tion has  the  inconvenience  of  throwing  the  face  whose 
indices  are  all  positive  into  the  negative  form  ;  but  as  it  is 


Systematic  Mineralogy.  [CHAP.  VII. 

that  most  generally  followed,  it  will  be  adopted  in  the  follow- 
ing pages.     The  notation  of  the  unit  forms  is  therefore  : 


1  1  1  |     n  1  1  1      i 

-  P= 


Ill  IIIIIII 

Weiss's  symbols  are  (a  :  b  :  c)  and  (a1  :  b  '.  c)  respectively. 

From  the  unit  hemipyramids  ±_  P  we  may,  by  keeping 
a  and  b  constant  and  altering  the  value  of  c,  obtain  other 
forms,  which  will  be  flatter  or  steeper  according  as  the  values 
assigned  are  greater  or  less  than  unity.  These  are  the  hemi- 
pyramids of  the  principal  series  ;  the  forms  are  represented 
by  the  symbols  : 

-  ±  P,  ( a  :  b  :  -  :  A  or  \h h  1}  where  (h<l\ 
m  V  m      J 

and  the  acuter  ones  by 

±mP,(a  :b  :  me]  or  {h  h  1}  where  (h  >  /). 

When  in  the  forms  ±  P  the  axes  b  and  c  are  kept  constant, 
and  a  is  lengthened,  a  new  series  known  as  clinodiagonal 
hemipyramids  are  obtained,  represented  by  the  symbols  : 

±  Pny  (na  :  b  :  c},  or  \hkl\  where  (h  <  k\ 

Here  the  oblique  axis  is  indicated  by  the  inclined  bar  in 
the  letter  P.  The  same  system  of  derivation  applies,  how- 
ever, to  any  of  the  forms  of  the  principal  series,  ±  P,  so  that 
the  general  symbols  of  any  hemipyramid  of  the  clinodiagonal 
series  are  : 

±  m  Pn,  (na  :  b  :  me)  or  {h  k  /} 

A  third  series,  as  in  the  rhombic  system,  is  obtained  by 
varying  the  length  of  the  orthodiagonal,  or  axis  of  symmetry  b, 
a  and  c  being  unchanged.  These  are  the  hemipyramids  of 
the  orthodiagonal  series,  whose  symbols  are  for  the  forms 
derived  from  +  P : 

±  JPn,  (a  :  nb  :  c\  or  {hkh\  where  (h  >  k). 


CHAP,  vil.]  Oblique  Prism.  149 

and  for  those  derived  from  any  other  of  the  forms  +  m  P  \ 
±  m  fn,  (a  \  nb  :  me)  or  {h  k  /} 

where  the  straight  bar  and  the  stem  of  the  P  signifies  that 
the  orthodiagonal  is  the  axis  modified.  The  geometrical 
character  of  these  forms  will  be  generally  analogous  to  those 
similarly  produced  in  the  rhombic  system,  the  difference 
between  the  two  hemipyramids  being  remembered.  When, 
however,  m  =  GO  or  /  =  o,  the  edge  lying  in  the  plane  of 
symmetry  becomes  vertical  or  parallel  to  c  for  either  hemi- 
pyramid,  and  a  prism  is  produced,  which  is  known  as  the 
primary  vertical  prism,  with  the  symbols  oo  P,  (a  :  b  '.  <x>c), 
and  {no.}  This  is  only  distinguishable  from  a  rhombic 
prism  by  the  circumstance  that  the  diagonals  of  the  rhomb 
forming  its  horizontal  section  are  not  the  axes  b  and  a,  but 
the  orthodiagonal  and  the  horizontal  projection  of  the  clino- 
diagonal,  and  therefore  the  fundamental  ratio  of  the  axes 
lying  in  the  basal  section  cannot,  as  in  the  rhombic  sys- 
tem, be  determined  from  a  measurement  of  the  angle  of 
the  prism  alone,  a  knowledge  of  the  characteristic  angle  /3 
being  required  in  addition.  From  the  clinodiagonal  hemi- 
pyramid  m  fn  by  a  similar  method  clinodiagonal  prisms, 
QO  fn  =  (na  :  b  :  <x>c}  =  [h  k  o}  where  (h  <  k]  are  derived, 
and  from  the  orthodiagonal  series  m  P  n,  orthodiagonal 
prisms  oo  Fn  =  (a  :  nb  :  cce)  =  {//  k  o}  where  (h  >  k).  The 
first  of  these  in  combination  modify  the  right  and  left  edges 
of  the  primary  prism,  and  the  second  those  in  the  front  and 
back  plane,  the  angles  being  more  obtuse  as  the  value  of  n 
increases. 

In  the  series  of  clinodiagonal  hemipyramids,  +  fn,  the 
angle  between  the  edges  in  the  plane  of  symmetry  and  the 
clinodiagonal  axis  becomes  more  acute  as  the  value  of  n  is 
increased,  and  when  it  becomes  GO,  the  four  faces  of  either 
hemipyramid  become  parallel,  forming  an  inclined  rhombic 
prism,  whose  edges  are  parallel  to  that  axis.  This  is 


150  Systematic  Mineralogy.  [CHAP.  VII. 

known  as  the  principal  clinodome  fee  =  (ao  a  :  b  :  c}=  (01  1}  . 
In  like  manner  a  more  obtuse  hemipyramid  +  —  f  gives 

rise  to  a  flatter  clinodome  —  f<x>=  (GO  a  :  b  :  —  c  }  = 

m  \  m      ) 

{o  k  /}  where  (k  <  /),  and  a  more  acute  one  ±  m  P  to  the 
steeper  form  m  f.cc=  (<x>a  :  b  :  me}  =  (ok  I)  where  (k  <  /). 
In  combination  the  nrst  of  these  modifies  the  top  and 
bottom  edges,  and  the  second  the  right  and  left  ones  of  the 
principal  clinodome  {on}. 

In  the  orthodiagonal  series  of  hemipyramids  the  angle 
made  by  the  faces  with  the  plane  of  symmetry  increases  with 
m,  becoming  90°  when  the  latter  =  <x>,  i.e.,  the  two  faces  of 
a  hemipyramid  right  and  left  of  the  plane  of  symmetry  fall 
into  one,  forming  with  their  corresponding  counterplanes 
pairs  of  planes  parallel  to  the  orthodiagonal  axis,  distinguished 
as  positive  and  negative  hemidomes,  in  the  same  way  as  the 
hemipyramids  whence  they  originate.  The  symbols  of  the 
different  series  of  these  forms  are  : 

Positive  hemidomes, 
Principal  J          +      f  oo  =  (a'  :  oo  b  :  c}  =  {i~oi}  . 

Flatter  forms       .    i    D          /  /         ,      i      "N      fr    « 
/;,  ^  7\          +  -  ^°°=  {«  :  oo  <*  :  -c  )  =  {//o/}. 
(n  <l)  m  \  m     )      l       ' 

Steeper  forms     +  m  fee  =(a'  :  cob  :  m  c)  =  {7i  o  /}  . 

(A  >  /) 

Negative  hemidomes. 

Principal  '  —      ^oo  =  (a  :  oo  b  :  c]  =  (101]  . 

Flatter  forms  i  „          /  ,      i     \       ,  . 

it,  *  t\  —~f^—(a  :  ccl>  :  -c  )  =  {//o/  . 

("<*.)  m  \  m     J      l 


Steeper  forms     —  m  fee  =  (a  :  oo  b  :  mc}={kol\. 
(*>/) 

1  In  these,  as  in  the  hemipyramids,  the  sign  of  the  first  index  in 
Miller's  symbol  is  the  reverse  of  that  of  Naumann's  symbol,  being 
negative  in  +  P  and  positive  in  -  P.  This  is  a  necessary  conse- 
quence of  expressing  the  obliquity  of  the  axes  by  the  acute  angle  0,  but 
if  the  obtuse  angle  is  used  the  type  face  of  +  P  becomes  (I  I  i),  or 
both  are  positive.  This  method  of  notation  is  adopted  by  Schrauf  in 
his  great  atlas  of  crystallography. 


CHAP.  VII.]  Oblique  Pinakoids.  151 

In  combination  the  principal  hemi-orthodome  truncates 
the  edges  in  the  plane  of  symmetry  of  the  corresponding 
hemipyramid  of  the  same  sign,  and  has  its  own  edges  of 
combination  bevelled  obliquely,  above  by  the  flatter,  and 
below  by  the  steeper  forms  of  the  same  class. 

Pinakoids.  When  the  inclined  axis  in  the  clinoprism 
and  the  vertical  axis  in  the  clinodome  are  made  infinitely 
long,  i.e.,  when  GO  is  substituted  for  m  and  n  in  the  symbols 
of  these  forms,  they  are  reduced  to  a  pair  of  planes  parallel 
to  the  front  and  back  axial  plane  or  plane  of  symmetry. 
This  is  known  as  the  clinopinakoid,  and  has  the  symbols 
GO  JsPoo  =  (oo  a  :  b  :  GO  c]  =  {010} .  It  truncates  all  prismatic 
and  domatic  edges  that  lie  in  the  orthodiagonal  or  transverse 
axial  plane. 

The  most  obtuse  kind  of  orthoprism  GO  fn  in  like  manner 
is  that  having  n  =  GO  or  GO  f<x>  =  (a  :  oo  b  :  GO  c)  =  {100} . 
This,  known  as  the  orthopinakoid,  is  parallel  to  the  right 
and  left  axial  plane,  and  truncates  the  edges  of  all  prisms 
lying  in  the  plane  of  symmetry.  The  third,  or  basal  pina- 
koid,  parallel  to  the  plane  containing  the  ortho-  and  clino- 
diagonal  axes,  has  the  symbols,  o  P  =  (GO  a  :  GO  b  :  c) 
=•(101-).  It  is  derived  from  all  hemipyramids,  hemi- 
domes,  and  clinodomes  when  o  is  substituted  for  m  in  their 
symbols. 

The  solid  formed  by  the  combination  of  the  three  pina- 
koids  having  its  edges  parallel  to  the  axes  represents,  when 
the  latter  are  developed  in  the  ratio  a  :  b  :  c  of  the  unit 
form,  the  fundamental  or  reticular  polyhedron  of  the  Bra- 
vais  notation,  the  molecular  meshes  in  the  clinopinakoid 
being  rhomboids,  whose  sides  are  in  the  proportion  a  :  c ; 
while  in  the  other  two  planes  they  are  rectangular,  having 
sides  in  the  proportion  of  a  :  b  and  b  :  c  respectively.  The 
angle  /3  is  directly  obtainable  from  this  combination  by 
measuring  the  inclination  of  the  face  (ooi)  upon  (100),  but  no 
other  element  can  be  determined  from  it,  all  the  other  angles 
being  right  angles.  The  appearance  of  crystals  belonging 
to  the  oblique  system  varies  very  considerably,  and  is  chiefly 


152  Systematic  Mineralogy.  [CHAP.  vil. 

determined  by  the  obliquity  of  the  axes  a  and  c  or  the 
angle  (3,  which  in  some  cases  is  as  low  as  55°  or  60°,  while 
in  others  it  may  only  differ  from  a  right  angle  by  a  few 
minutes,  and  even  this  difference  is  not  essential,  as  there 
are  instances  of  minerals  with  the  three  axes  at  right  angles 
to  each  other  having  the  symmetry  and  physical  properties 
characteristic  of  this  system.  The  determination  of  such 
forms  depends,  however,  not  so  much  on  considerations  of 
shape  as  on  physical  properties,  as  will  be  subsequently 
shown ;  but  the  general  principle  may  be  laid  down  that 
the  essential  characteristic  of  the  system  is  not  an  oblique 
axis,  but  monosymmetry,  and  therefore  the  name  mono- 
symmetric  is  more  generally  appropriate  for  the  system, than 
oblique  or  monoclinic,  and  would  be  preferable  were  not 
these  latter  terms  of  much  wider  currency. 

As  only  one  of  the  principal  crystallographic  lines  coin- 
cides with  an  axis  or  direction  of  physical  equivalence  in 
oblique  crystals,  any  position  that  satisfies  the  general  sym- 
metry may  be  adopted  for  reading  them.  It  is,  however,  prac- 
tically inconvenient  to  adopt  faces  as  base  and  orthopinakoid 
that  make  a  very  obtuse  angle  with  each  other,  as  the  resulting 
axial  obliquity  ft  and  one  of  the  angles  of  the  triangles  arising 
in  calculation  will  be  so  acute  that  the  sides  cannot  be 
computed  with  accuracy,  as  a  small  variation  in  the  angle 
produces  a  great  difference  in  the  length  of  the  side  opposite 
to  it.  A  complete  determination  of  the  elements  of  the 
crystal  is  only  possible  when,  in  addition  to  two  pairs  of 
faces  perpendicular  to  the  plane  of  symmetry,  which  are 
considered  as  oP  and  ccfao,  either  one  hemipyramid  or 
two  prismatic  forms  belonging  to  different  zones,  such  as 
GO  P  and  ^*GO  are  present.  From  a  combination  containing 
only  a  prismatic  form  with  oblique  end  faces,  the  ratio  a  :  b 
and  the  angle  ft  may  be  determined,  but  not  the  length  of 
the  vertical  axis.  The  appearance  of  monosymmetric  com- 
binations varies  with  their  crystallographic  elements.  When 
the  obliquity  of  the  axis  a  is  considerable,  and  the  faces  are 


CHAP.  VII.] 


Oblique  Combinations. 


153 


numerous,  the  difference  from  those  of  systems  whose  sym- 
metry is  more  complete  is  very  marked  ;  while  on  the  other 
hand,  those  having  a  nearly  horizontal  clinodiagonal  axis,  and 


FIG.  238. 


FIG.  239. 


few  or  slightly  developed  faces  belonging  to  hemipyramids  or 
hemidomes,  are  often  scarcely  distinguishable  from  rhombic 
or  hexagonal  forms.  The  leading  test  is,  however,  in  all  cases 


FIG.  240. 


FIG.  241. 


01° 


*\^^                                  OOJ                                 ^^ 

1U1 

Ho 

100 

no 

-    / 

10! 

the  presence  of  similar  faces  right  and  left  of  the  plane  of 
symmetry,  combined  with  differences  in  the  faces  modifying 
the  top  and  bottom  edges  parallel  to  the  axis  b.  Some  of 
the  simpler  cases  are  represented  in  figs.  238-247.  Fig. 


1 54  Systematic  Mineralogy.  [CHAP.  VII. 

238  has  the  front  lower  and  back  upper  solid  angles  in  the 
plane  of  symmetry  in  the  combination  oo  P,  o  P,  modified  by 
an  acute  hemidome  +  3  ^*> ;  while  in  fig.  239  the  same 


FIG.  242. 


FIG.  243. 


class  of  combination  has  in  addition  a  less  acute  hemidome 
—  2  fee  at  the  top  in  front  and  the  orthopinakoid  GO  fee. 
Both  are  forms  of  Clinoclase.  Fig.  240,  a  crystal  of  Brushite, 


FIG.  244. 


has  in  addition  to  the  plane  of  symmetry  oo  ^oo,  the  basal 
pinakoid  o  P,  a  hemi-orthopyramid  +  f  2,  and  an  ortho- 
prism  00^*2.  Fig.  241,  a  combination  of  a  nearly  rhombic 
character  observed  in  Caledonite,  is,  however,  distinguishable 


CHAP.  VII.] 


Oblique  Combinations. 


155 


by  the  hemi-orthodome  +  \  fee,  which  appears  under 
the  unit  hemi-orthodome  +  ^oo  in  front  below,  but  not 
above.  Fig.  242,  the  common  form  of  Borax  crystal,  con- 
tains the  three  pinakoids,  the  principal  prism,  and  both  prin- 
cipal hem ipyramids,  but  the  oblique  character  is  brought  out 
by  the  acute  hemipyramid  2  P,  which  occurs  only  in  the 
positive  form.  Fig.  243,  a  crystal  of  Allanite,  contains  ccP, 
GO  Pec,  o  P,  and  +  ^oo.  Fig.  244,  one  of  the  simplest  forms 


FIG.  246. 


FIG.  247. 


110 


J 


of  Hornblende,  contains  only  trre^anit  prism  <x>  P  and  clino- 
dome  fee  ;  fig.  245  the  same,  with  the  acute  edges  of  the 
prism  truncated  by  the  clinopinakoid  ccPcc.  Fig.  246  has 
the  edges  between  oo  P  and  o  P  modified  by  the  faces  of  the 
negative  hemipyramid  —  P,  together  with  the  clinopinakoid 
ccPcc.  Fig.  247,  a  crystal  of  Azurite,  elongated  parallel 
to  the  orthodiagonal  axis,  contains  o  P,  —  ^  P  oo,  oo  Pec, 
+  j  Pec,  and  +  jjp2.  These  will  be  sufficient  to  indicate 
the  general  character  of  the  combinations  of  this  system. 
A  few  more  complex  examples  will  be  given  in  the  descrip- 
tive portion  of  the  work. 


I56 


Systematic  Mineralogy.          [CHAP.  VIII. 


CHAPTER  VIII. 

TRICLINIC1     SYSTEM. 

THE  forms  of  this  system  are  referred  to  three  axes  all 
having  different  parameters  and  all  oblique  to  each  other. 
The  characteristic  elements  of  crystals  belonging  to  it  are 
therefore,  in  addition  to  the  lengths  cf  the  axes,  the  three 
angles  between  them.  This  gives  forms  of  the  most  rudi- 
mentary character,  every  face  crystallographically  possible 
— that  is,  having  a  similar  face  parallel  to  itself  as  required 
by  the  general  conditions  of  crystallographic  symmetry  to  a 
centre — is  a  complete  form,  and  may  combine  with  any 
other  having  similar  or  dissimilar  indices ;  and  as  no  form 
can  have  more  than  two  faces,  any  actual  crystal  must  be  a 
combination  of  at  least  three  forms.  From  the  obliquity  of 
the  axes  there  can  be  neither  planes  nor  axes  of  symmetry, 
which  property  is  indicated  in  the  name  '  asymmetric.' 

The  notation  of  the  axes  is  similar  to  that  in  the  rhombic 
system,  when  one  has  been  selected  as  the  vertical  axis  c, 


the  shorter  one  of  the  other  two  is  made  the  brachydiagonal 
and  the  longer  the  macrodiagonal,  the  latter  being  so 
arranged  as  to  slope  from  left  to  right.  The  angle  between 

1  Other  names  are  anorthic,  asymmetric,  doubly- oblique,  oblique- 
rhomboidal,  and  eingliedrig. 


CHAP.  VIII.] 


Triclinic  Symmetry. 


157 


c  and  b  is  called  a,  that  between  c  and  a,  /3,  as  in  the  oblique 
system,  and  that  between  a  and  b,  y  ;  these  angles  being 
measured  upon  the  positive  semi-axes,  as  shown  in  per- 
spective projection  fig.  248,  and  in  the  orthographic  projec- 
tions upon  the  three  principal  sections  (figs.  249,  250,  and 


251).  Each  of  these  latter  shows  two  of  the  semi-axes  in 
their  true  lengths  and  inclinations,  the  notation  being  gene- 
rally similar  to  that  of  the  rhombic  system.  From  these  it 
will  be  seen  that  the  principal  sections  are  all  rhomboids, 


and  that  the  particular  solid  corresponding  to  three  finite 
parameters  is  an  oblique  rhomboidal  pyramid  or  octahe- 
dron contained  by  four  dissimilar  pairs  of  faces,  each  of 
which,  therefore,  represents  a  different  form.  These  are 
known  as  quarter-  or  tetarto-pyramids,  whose  positions  are 


158  Systematic  Mineralogy.          [CHAP.  VIII, 

in  Naumann's  method  indicated  by  the  letter  P  differ- 
ently accented,  according  as  the  face  indicated  belongs  to 
the  right  or  left,  upper  or  lower  octants.  The  complete 
notation  according  to  the  different  systems  is  as  follows  : — 

I.  (a  :  b  :  ')=(][.  ££)}/>'      H-  (a>  '•  b  '•  <:)=(I  llH  p 
VII.  (a!  :  V  :  c'}—(\  i  i)l       VIII.  (a  \  V  \  t')=(i  i  i)l' 

III.  (a1  :  b1  :  *)=(!!  i))  P      Iy.  (a  :  b'  :  c)=(i ~i 
V.  (a  :  b  :  O=(i  i  ?)'   •       VI.  (a'  :  b  :  O=(i  i 

The  relations  of  the  observed  forms  may  be  developed 
from  the  symbols  of  any  of  the  unit  quarter-pyramids  in  a 
similar  manner  to  that  given  for  the  rhombic  and  oblique 
system.  Thus,  by  varying  the  length  of  c,  while  a  and  b  are 
unchanged,  P'  gives  rise  to 

m  P'  =  (a  :  b  :  mc}-=  \Jih  /}  where  h  >  / 
when  c  is  lengthened,  and  to 

1  P<  =  (a  :  b  :  -  c]  =  {hhl}  where  h<l 
m  V,  «  x 

when  the  vertical  axis  is  shortened. 

In  like  manner  a  similar  group  is  derived  from  each  of 
the  three  other  quarter-pyramids,  which,  together  with  the 
preceding,  form  the  principal  or  vertical  series. 

By  altering  the  length  of  the  right-and-left  axis  b,  in 
the  principal  form  m  P',  a  and  c  being  unchanged,  a  macro- 
diagonal  series  of  quarter-pyramids  arises,  having  the 
symbols : 

m  P'  n  =  (a  :  nb  :  me}—  {hkl}, 

and  when  the  axis  a  is  similarly  varied  the  result  is  the 
brachydiagonal  series  having  the  symbols  : 

m  P'  n  =  (n  a  :  b  :  m  c)  =  {hkl}. 

In  all  these  series  of  derived  quarter-pyramids,  when  m 
becomes  oo,  the  form  changes  to  a  pair  of  planes  parallel  to 
the  vertical  axis,  or  a  herniprism,  which  includes  both  the 


CHAP.  VIII.]  Triclinic  Hemiprisms.  159 

upper  and  lower  quarter-pyramids  lying  on  the  same  side  of 
the  centre.  This  is  indicated  by  the  position  of  the  accents 
in  Naumann's  symbol.  Thus,  from  P'  and  P,  is  derived 
the  right  principal  hemiprism  : 

oo  P/  =  (a  :.  b  :  00  c]  =  {i  i  o} , 
and  from  'P  and  tP,  the  left  principal  hemiprism, 
ao/P=(a  :  V  :  oo^)  =  |ilo}. 

.which,  like  all  other  triclinic  forms,  may  appear  together 
in  the  same  combination,  or  independently  of  each  other, 
there  being  no  true  triclinic  prism,  but  only  a  prismatic 
combination  of  the  hemiprisms.  By  increasing  the  length 
of  the  macrodiagonal  or  brachy diagonal  axis  respectively  in 
the  principal  hemiprisms,  the  other  axes  being  unchanged, 
macrodiagonal  and  brachydiagonal  hemiprisms  are  formed. 
The  symbols  of  the  former  are 

oo  PI  n.  oo  jPn  =  (a  :  n  b  :  oo  c]  (a  \  n  b1  :  oo  c}  = 
{h  k  o}   {h  k  o}  where  h  >  k, 

and  those  of  the  latter 

oo  Pi  n,  oo  /Pn  =  (n  a  :  b  :  oo  c)  (n  a  :  b1  :  oo  c)  = 
{//  k  o}   \h  k  o}  where  h  <  k. 

In  the  macrodiagonal  quarter-pyramids,  P'  n  and  'Pn,  when 
n  =  oo,  the  angle  between  the  faces  meeting  in  the  front 
and  back  axial  plane  becomes  180°,  or  two  fall  into  one 
parallel  to  the  macrodiagonal,  producing  a  hemi-macrodome, 
'.Poo  —  (a  :  cob  :^)  =  {ioi},  which  in  combination  trun- 
cates edges  parallel  to  that  axis  in  front  of  the  crystal  above 
the  base  and  below  it  behind,  the  correlated  form^oo, 
having  the  reverse  position,  or  appearing  below  in  front 
and  above  behind.  Other  analogous  forms  represented  by 
m  ' P  oo  and  m  ,P,  oo  are  derived  in  the  same  way  from  the 
quarter-pyramids  m  Pn,  m  'Pn,  &c. 

Hemiprismatic  forms  parallel  to  the  axis  a,  or  hemi- 
brachydomes,  are  obtained  from  pairs  of  the  quarter-brachy- 


160  Systematic  Mineralogy.          [CHAP.  VIII. 

pyramid  series  by  making  n  =  GO  in  their  symbols,  when  the 
two  faces  meeting  in  the  right  and  left  axial  plane  fall  into 
one.  That  derived  from  P1  n  and  ,Pn  has  the  symbols 

,P  GO  =  (oo  a  :  b  :  c}  =  {o  i  1} 

which  in  combination  modifies  edges  parallel  to  the  brachy- 
diagonal  axis  above  the  centre  of  the  crystal  to  the  right 
and  below  it  to  the  left,  while  the  correlated  form  derived 
from 

'Pn  and  P,  n  or  'P,  oo  =  (oo  a  :  b'  :  c]  =  {o"i  1} 

has  the  opposite  position  or  to  the  right  below  and  to  the  left 
above.  As  before,  the  general  symbols  for  the  hemi-brachy- 
domes  are 

m  ,P  GO  =  (oo  a  :  b  :  m  c)  =  (o  k  1} 
and 

*«  './*,  GO  =  (GO  0  :  b'  :  m  c )  =  {o  /£  /} . 

In  these,  unlike  the  other  prismatic  forms  (the  hemiprisms 
and  hemi  macrodomes),  the  two  quarter-pyramid  planes  in- 
cluded in  any  face  have  dissimilar  signs,  or  one  is  a  type  or 
positive  plane  of  one  form,  and  the  other  the  negative  or 
counter-plane  of  another.  Hence  the  accents  in  Naumann's 
symbols  lie  chequerwise,  as  in  naming  the  forms  the  front 
planes  are  always  meant 

In  the  hemi-brachydomes,  m  ,P'  GO  and  m 'P,  GO,  when  m 
is  made  =  GO,  the  faces  become  parallel  to  the  front  and 
back  axial  plane,  producing  the  brachypinakoid,  which,  like 
that  in  the  rhombic  system,  has  the  symbols 

GO  ^oo  =  (GO  a  :  b  .  GO  c)  •-=  {o  i  o} 

and  similarly  m  =  GO  in  a  hemi-macrodome  gives  rise  to  the 
form  parallel  to  the  right  and  left  axial  plane  or  the  macro- 
pinakoid 

GO  ^oo  =  (a  :  oc£  :  'GO  c)  =  (i  o  o). 
The  third,  or  basal  pinakoid,  is  the  limiting  form  of  the 


CHAP.  VIII.]  Triclinic  Combinations.  161 

vertical  series  of  quarter-pyramids  n'  tF-  when  ;//  =  o,  and 
is  represented  by 


This  system  of  development  may  be  represented  by  the 
scheme  given  for  the  rhombic  system  on  p.  134,  if  the 
octants  in  which  the  particular  quarter-  pyramids  and  hemi- 
prismatic  forms  lie,  be  indicated  by  properly  accentuating 
the  symbol  P.  The  lines  on  which  the  symbols  are  arranged 
will  also  have  the  same  significance,  that  is,  those  in  any 
horizontal  and  vertical  lines  will  be  in  the  corresponding 
zones,  except  that  the  axes  of  the  principal  zones,  instead 
of  being  at  right  angles,  will  be  oblique  to  each  other. 

The  appearance  of  triclinic  combinations  is  chiefly  de- 
pendent upon  the  obliquity  of  the  axes.  When  the  three 
angles  differ  but  slightly  from  right  angles,  as  in  the  mineral 
Cryolite,  the  crystals  have  a  general  resemblance  to  cubic 
forms,  while,  on  the  other  hand,  in  Axinite  and  Sulphate  of 
Copper,  they  are  marked  by  extreme  obliquity  and  apparent 
want  of  symmetry.  In  other  species,  notably  in  the  felspar 
group,  triclinic  crystals  occur,  which  in  Albite  are  closely 
allied  morphologically  to  those  of  the  analogous  species 
Orthoclase,  in  the  oblique  system.  In  this  latter  case  the 
resemblance  is  often  so  close  that  the  system  to  which  the 
crystals  belong  cannot  always  be  determined  by  considera- 
tion of  forms  alone.  The  combination  of  the  three  pina- 
koid  planes,  also  called  the  doubly-oblique  prism,  is  the 
primitive  solid  of  the  system  according  to  the  French  nota- 
tion, the  faces  being  oblique  parallelograms  whose  sides 
represent  the  meshes  of  the  molecular  network,  each  being 
dissimilar  from  the  other  two. 

The  determination  of  the  elements  of  a  triclinic  form 
requires  at  least  five  independent  observations,  and  involves 
calculations  which  cannot  be  described  in  few  words.  The 
student  is  therefore  referred  for  information  on  this  subject 
to  the  larger  works  on  determinative  mineralogy  and  prac- 
tical crystallography.  As  there  is  no  direct  relation  between 

M 


1 62 


Systematic  Mineralogy,         [CHAP.  VIII. 


form  and  other  physical  properties,  the  choice  of  position  is 
quite  arbitrary,  so  that  there  may  be  and  often  is  consider- 
able diversity  of  opinion  as  to  the  symbols  to  be  assigned  to 
the  faces  by  different  authors. 

The  general  characters  of  the  simpler  triclinic  com- 
binations will  be  seen  in  figs.  252-254.  Fig.  252  is  one  of 
the  most  un  symmetrical  kinds,  a  crystal  of  Axinite,1  contain- 
ing 'P.  P'.  \  P1.  o  P'.  'P'  oo.  2  'P'  oo. 


FIG.  252. 


FIG.  253. 


Fig-   253>   a  crystal  of  Babingtonite,  contains  oo^Poo. 
oo  Poo.  oP.  oo-'jPf.  ^'oo.  'J^QO.     Fig.  254  is  similar,  with 

FIG.  254-  Fin.  255. 


the  substitution  of  the  hemiprisms,  <x>P'.  ocLP2.  Fig.  255, 
a  crystal  of  Albite,  has  a  general  resemblance  to  one  oi 
Orthoclase,  but  the  special  triclinic  character  is  apparent  by 
the  presence  of  the  quarter-pyramid  P,  only  on  the  edges 
between  GO  ^oo  and  ,P,  oo  and  not  on  the  analogous  edges 
between  GO  ^co  and  o  P. 

1  This  is  the  position  adopted  by  Schrauf.     Other  authors  consider 
the  quarter-pyramid  faces  as  belonging  to  the  zone  of  the  prism. 


CHAP.  IX.]  HcmimorpJiic  Crystals.  163 


CHAPTER  IX. 

COMPOUND   OR   MULTIPLE   CRYSTALS. 

IN  demonstrating  the  geometrical  characteristics  of  the 
different  systems  in  the  preceding  chapters,  the  solids  illus- 
trated have  been  assumed  to  be  of  the  most  regular  cha- 
racter, every  face  of  the  same  form  being  similarly  placed  in 
regard  to  the  symmetrical  centre  or  origin  of  the  axes. 
Such  crystals,  however,  without  being  absolutely  unknown, 
are  of  comparative  rarity,  at  any  rate  in  individuals  of  any 
size,  and  in  by  far  the  larger  number  of  instances  one  or  more 
faces  of  any  form  may  be  largely  developed,  with  a  corre- 
sponding reduction  or  even  entire  suppression  of  the  re- 
mainder, as,  for  example,  in  the  common  case  of  a  prism 
terminated  by  pyramids  or  domes,  the  faces  of  the  latter 
forms  appear  only  at  one  end  of  the  prism,  because  the 
other  forms  the  surface  of  attachment  to  the  rock.  In  such 
cases  the  missing  faces  have  to  be  assumed  in  reasoning 
out  the  character  of  the  completed  from  the  observed  form. 
Besides  this,  there  are  other  cases  in  which  two  or  more 
crystals  are  united  into  a  mass  having  a  particular  regularity 
of  arrangement,  the  component  crystals  preserving,  to  a 
great  extent,  their  individuality.  Such  multiple  crystals  are 
of  two  principal  kinds  known  as  parallel  and  twinned  groups, 
but  before  considering  these  it  is  necessary  to  notice  a  third 
special  kind  of  development  which  is,  to  some  extent,  of  a 
compound  character. 

Hemimorphism.  There  are  a  few  minerals  ana  artificial 
products,  whose  crystals  are  dissimilarly  ended,  the  faces 
limiting  a  prismatic  zone  at  one  end  of  its  axis  belonging  to 
different  forms  from  those  in  the  corresponding  position  at 
the  other  end.  Such  crystals  are  not  properly  hemihedral, 
as,  although  they  contain  but  half  the  full  number  of  faces 
possible  in  their  constituent  forms,  these  faces  are  not,  as 


164 


Systematic  ^Mineralogy. 


[CHAP.  IX. 


FIG.  256. 


they  should  be,  uniformly  distributed  about  the  axes,  but 
are  so  grouped  that  we  may  have  all  the  faces  whose  indices 
are  positive  to  an  axis,  while  the  corresponding  negative 
ones  are  entirely  absent,  their  places  being  occupied  by 
some  totally  different  form.  This  arrangement  is  incom- 
patible with  regular  hemihedrism,1  and  it  is  therefore 
distinguished  by  the  name  of  hemimorphism.  The  most 
conspicuous  examples  are  afforded  by  Tourmaline,  the  Ruby, 
Silver  Ores,  and  Greenockite  in  the  hexagonal,  Struvite  and 
Electric  Calamine  in  the  rhombic,  and  Gane  Sugar  in  the 
oblique  system. 

Fig.  256  represents  a  crystal  of  Tourmaline  contained 
above  by  R  v  {i  o  1 1}  (a),  and  —  R  K  (o  i  T  1}  (b) ;  below,  by 
R(a)  and  —  ^R  K  (i_p  fz}  (c) ;  and  in  the  zone  of  the 
prism  by  ccPz  {1120}  (d\  <x>P  {o  i  i  o}  (<?),  and  <x>P$ 
{1340}  (/).  Of  the  latter  three  forms  the  first  appears  with 
its  full  number  of  faces,  and  the  others 
with  only  one-half,  as  trigonal  and  di- 
trigonal  prisms  respectively.  The 
reason  of  this  is,  that  the  prism  of 
the  first  order,  considered  as  a  rhom- 
bohedron  of  infinite  altitude,  falls 
into  two  groups  of  three  faces,  one 
of  which  belongs  to  the  upper  and 
the  other  to  the  lower  end  of  the 
crystal,  either  of  which  may  be  pre- 
sent to  the  exclusion  of  the  other  in 
a  hemimorphic  group,  and  the  dihex- 
agonal  prism  in  the  same  way  as  an 

unlimited  scalenohedron  divides  into  an  upper  and  a  lower 
ditrigonal  prism  ;  but  a  face  of  the  hexagonal  prism  of  the 
second  order  includes  both  upper  and  lower  faces  of  the  sca- 
lenohedron, of  which  either  maybe  omitted  without  changing 
its  geometrical  character.  The  occurrence  of  trigonal  prisms  of 

1  This  is,  however,  distinguished  by  Miiller  as   asymmetric  hemi- 
hedrism  in  the  rhombqhedral  system. 


CHAP.  IX.] 


Hemimorphic  Crystals. 


this  kind  is  therefore  evidence  of  hemimorphic  development, 
even  when  both  ends  of  the  crystal  are  not  available  for 
observation,  as  is  generally  the  case  in  the  ruby  silver  ores. 
In  Greenockite  the  dissimilarity  of  the  ends  is  much  more 
marked,  the  crystals  showing  numerous  pyramids  at  one 
end  of  the  prism,  squared  off  by  the  basal  plane  at  the 
other.  Fig.  257,  a  crystal  of  Struvite,  has  the  upper  faces  of 
Poo  {i  o  i},  combined  with  the  lower  ones  of  J^oo  {103} 
and  o  P.  (o  o  1} ,  which  are  limited  transversely  by  P<x>  {o  1 1 } , 
4/*oo  {041},  and  <x>P<x>  {oio};  the  latter  may  be  con- 


FIG.  257. 


FIG.  258. 


sidered  as  common  to  both  sides  of  the  base,  while  the  for- 
mer two  are  only  represented  by  their  upper  faces.  Fig.  258  is 
a  crystal  of  Electric  Calamine  contained  by  <x>P{i  i  o}, 
oo  fee  {i  o  o} ,  oo  Pcc  {o  i  o} ,  limited  above  by  3  Poo  (301), 
3^00  {031},  and  o/>{ooi},  and  below  by  the  brachy- 
pyramid  2  p2  {12 1).  These  are  some  of  the  more 
striking  examples  of  this  class  of  crystals,  which,  as 
a  rule,  are  distinguished  by  the  property  of  pyroelec- 
tricity,  the  opposite  developing  dissimilar  polarity  when 
heated. 

Parallel  grouping.  In  the  simplest  case  of  the  aggre- 
gation of  two  similar  crystals,  the  individuals  are  so  arranged 
that  a  line  joining  their  centres  is  either  on  the  prolongation 
of  a  crystallographic  axis  or  parallel  to  it,  as  in  fig.  259, 
representing  two  octahedra  having  a  common  vertical  axis 


i66 


Systematic  Mineralogy. 


[CHAP    IX. 


FIG.  259. 


where  the  surface  of  contact  represented  by  the  shaded 
plane  is  obviously  equivalent  to  a  face  of  a  cube,  and  no 

alteration  of  character  is  ef- 
fected by  mere  rotation  of 
either  crystal  through  one  or 
more  right  angles  about  the 
line  OTO'.  The  compound 
nature  of  such  growth  is  evi- 
denced by  the  re-entering 
angles  of  the  faces  adjacent  to 
the  plane  of  contact  which  will 
be  more  apparent  as  the  dis- 
tance between  the  centres 
O  O'  is  increased.  This  kind 
of  grouping,  often  many  times 
repeated,  is  commonly  seen  in 
crystals  of  alum,  and  also  in 
native  silver  and  other  cubic  minerals.  If  we  suppose  two 
cubes  to  be  united  in  the  same  way,  there  will  be  a  mere 

shifting  of  the  top  and 
bottom  faces,  giving  a 
cube  drawn  out  in  height, 
but  otherwise  indistin- 
guishable from  a  single 
crystal.  The  same  re- 
mark holds  good  if  either 
cubes  or  octahedra  are 
in  contact  parallel  to  a 
face  of  the  rhombic  do- 
decahedron, their  aspect 
not  being  changed  by  a 
half  turn  about  the  normal 
to  that  face. 

Twin  grouping.     Fig. 
260    represents    another 
method  of  contact  of  two  octahedra,  namely,  on  a  face  com- 


FIG.  260. 


CHAP.  IX.] 


Twin  Structure, 


167 


mon  to  both,  in  which  the  axes  are  parallel  as  long  as  the  par- 
ticular position  is  retained,  but  if  one  crystal,  say  the  front  one, 
be  turned  180°  about  the  line  0  TO',  the  result  shown  in 
fig.  261  is  obtained,  where  the  axes  are  no  longer  parallel, 
the  original  positive   extremities   in   the  movable   crystal 
coinciding  with  the_negative  ones  in  the  fixed  one,  or  A 
with  A',  £  with  £',  and  C  with  C,  while  their  opposite 
extremities  make  large  angles  with  each  other,  the  individual 
crystals  being  symmetrical 
to  their  common  face,  the 
surface  of  contact,  which, 
however,  as  we  have  pre- 
viously seen,  is  not  a  plane 
of  symmetry  of  the  form. 
This  arrangement  is  known 
as  a  twin  structure,  or  twin 
crystal,  the  common  plane 
of  symmetry  is   the  twin 
plane,  its  normal  the  twin 
axts^  and  the  surface  join- 
ing   the    two  crystals,  the 
plane    of  contact  or    com- 
position.      In   this,    as    in 
many  other  simple  cases,  the  planes  of  twinning  and  com- 
position coincide,  but  it  is  not  always  so,  and  the  distinc- 
tion between  them  must  be  carefully  borne  in  mind,  espe- 
cially in  dealing  with  the  twin  forms  in  the  systems  of  lower 
symmetry  where  only  the  composition  face  is  apparent,  and 
the  position  of  the  crystals  must  often  be  shifted  to  arrive  at 
the  true  twin  plane.     As  a  rule,  the  twin  plane  may  be  any 
actual  or  possible  face  of  a  form  proper  to  the  series,  other 
than  a  plane  of  symmetry,  and  it  is  generally  one  having 
low  indices,  such  as  110,100,  i  i  i,  &c.     If  we  suppose 
two  octahedra  in  the  position  of  fig.  261  to  be  freely  pene- 
trable, and  the  line  O  T  O'  to  be  shortened  until  O  and  O' 
coincide,  we  obtain  the  solid  fig.   262,  where  the  faces  of 


i68 


Systematic  Mineralogy. 


[CHAP.  IX. 


contact  lie  in  the  same  planes  in  front  and  behind,  but  all 
the  others  meet  in  re-entering  angles,  the  points  and  edges 
of  both  crystals  being  fully  developed.  This  is  known 
as  a  penetration  twin,  the  shaded  parts  belonging  to  the 
inverted,  and  the  white  to  the  direct  or  fixed  crystal,  the 
faces  being  numbered  according  to  the  original  positions  in 
the  preceding  figures. 

If  the  individual  crystals,   instead  of  being  regularly 
developed,  are  supposed  to  be  flattened  to  one  half  of  their 

FIG.  262. 


normal  thickness  upon  the  twin  axis,  the  group  will  resemble 
fig.  263,  where  there  is  no  penetration,  and  only  those 

faces  that  are  parallel  to  the 
twin  plane  appear  of  their  full 
size.  This  is  exactly  what  hap- 
pens when  a  single  crystal  is 
divided  by  a  twin  plane  pass- 
ing through  the  centre  as  in 
fig.  264,  and  one  half  turned 
through  1 80°,  the  other  re- 
maining stationary.  This  is  one 
of  the  most  convenient  methods 
of  explaining  twin  structure,  and 
is  that  most  generally  used,  the 
resulting  forms  are  called  contact-twins  as  well  as  modes  and 
liemtropc  crystals.  The  latter  terms,  which  were  formerly 


CHAP.  IX.] 


Cubic  Twin  Crystals. 


169 


in  general  use,  are  now  mainly  confined  to  the  works  of 
French  authors.  German  writers  describe  twin  crystals, 
zwilling,  drilling,  vielling,  &c.,  according  as  two,  three,  or 
more  individuals  are  apparent  in  the  group.  Fig.  265  is  a 
contact  twin  of  two  rhombic  dodecahedra  upon  a  face  of 
the  octahedron.  Here  there  are  no  re-entering  angles,  the 
section  upon  the  twin  plane  being  a  regular  hexagon.  A 


FIG.  265. 


FIG.  266. 


FIG.  267. 


complete  penetration  twin  of  the  same  kind  has  also  been 
observed  in  crystals  of  Sodalite.  Fig.  266  is  a  penetration 
twin  of  two  cubes,  which,  being  exactly  centred,  have  their 
points  on  the  twin  axis  in  common.  This  is  a  common 
twin  form  of  Fluorspar,  but  the 
observed  crystals  are  not  gene- 
rally quite  regular,  so  that  the 
projecting  portions  of  the  shaded 
crystal  above  the  faces  of  the 
white  one  are  alternately  of 
different  sizes  instead  of  being  all 
exactly  alike. 

The  greater  number  of  cases 
of  twin  crystals  among  holohedral 
cubic  forms  are  upon  the  above  type  where  the  twin  plane 
is  the  face  of  an  octahedron,  and  this  is  also  seen  in  inclined 
hemihedral  forms,  as  in  fig.  267,  a  penetration  twin  of  two 


Systematic  Mineralogy.  [CHAP.  IX. 

tetrahedra,  two  of  whose  faces,  parallel  to  the  twin  plane, 
lie  in  the  same  surface  at  one  end  of  the  twin  axis,  while  the 
other  six  meet  in  the  same  point  at  the  other  end. 

Fig.  268  is  a  more  common  case  of  penetration-twinning 
of  tetrahedra,  the  twin-plane  shown  by  oblique  shading 
being  a  face  of  the  cube.  When  the  individuals  in  such  a 

FIG.  268.  FIG.  269. 


group  instead  of  being  simple  tetrahedra  are  unequally 
developed  combinations  of  both  positive  and  negative  ones, 
the  appearance  is  similar  to  that  of  fig.  269,  or  an  octahedron 
with  a  V-^aped  groove  along  each  of  its  edges ;  and  the 

combination  of  -  QO  O  (fig.  95),  twinned  in  the  same  way, 
2 

resembles  a  rhombic  dodecahedron  grooved  parallel  to  the 
longer  diagonals  of  its  faces. 

In  the  parallel  hemihedral  forms  one  of  the  most  fre- 
quently observed  cases  is  the  penetration  twin  of  two  penta- 
gonal dodecahedra    — — -     (fig.  270),  the  twin  plane  being 
L.  2    J 

a  face  of  the  rhombic  dodecahedron.  This  is  especially 
characteristic  of  iron  pyrites,  and  the  similarly  constituted 
sulphides  and  arsenides  of  nickel  and  cobalt 

The  above  are  the  principal  kinds  of  twin-crystals  in  the 
cubic  system,  in  their  simplest  and  most  regular  develop- 
ment ;  other  and  more  complex  cases  arise  when  the  compo- 


CHAP.  IX.] 


Cubic  Twin  Crystals. 


171 


nent  crystals  are  of  different  sizes,  or  the  plane  of  composition 
is  not  central,  when  the  groups  are  often  considerably  dis- 
torted. 

The  same  structure  may  also  be  repeated  with  three  or 
more  individual  crystals,  producing  multiple  or  polysynthetic 

FIG.  271. 


twin  groups.  Fig.  271  is  an  example  of  a  peculiar  polysyn- 
thetic twin  of  Spinel  recently  described  by  Struver.  It  is 
made  up  of  six  tetrahedral  combinations,  o  t  to  o  6,  the  first 
four  being  repeated  contact  twins,  on  an  octahedral  face, 
while  the  fifth  and  sixth  are  parallel  to  the  second  and  third; 
and  as  all  their  twin  axes  lie  in  the  same  face  of  a  rhombic 
dodecahedron,  whose  axis,  the  line  joining  the  hollow  six- 
faced  angles  in  the  centre,  is  the  edge  common  to  all 
the  individuals,  there  is  complete  lateral  symmetry  to  that 
face. 

In  many  instances  the  structure  of  a  twin  group  may  be 
explained  in  more  than  one  way,  or  the  twin  axis  may  be 
exchanged  for  another  line  at  right  angles  to  itself,  rotation 
about  which  produces  a  similar  geometrical  form,  although 
the  position  of  individual  faces  may  be  different.  Thus,  in 
fig.  264  the  axis  O  T  may  be  exchanged  for  a  line  in  the 
twin  plane  joining  the  middle  points  of  opposite  edges,  and 
normal  to  a  new  plane,  cutting  the  edges  at  one-half  and 
one-third  of  their  lengths  alternately,  which  has  the  pro- 
perties of  a  face  of  the  icositetrahedron  2  O  2  and  gives 


Systematic  Mineralogy. 


[CHAP.  IX. 


forms  exactly  similar  to  figs.  262  and  263,  but  with  this 
difference,  that  the  faces  brought  opposite  to  each  other 
belong  to  different  tetrahedra,  instead  of  to  the  same  one  as 
they  do  with  the  octahedron  face  in  the  twinning  plane. 

Twin  crystals  of  the  hexagonal  system.  The  faces  of  di- 
hexagonal  pyramids  and  prisms  and  hexagonal  pyramids  are 
possible  twin  planes  in  the  holohedral  forms  of  this  system, 
but  the  only  observed  groups  are  twinned  upon  the  latter 
form,  and  they  are  not  of  very  common  occurrence.  In  the 
rhombohedral  hemihedral  forms,  on  the  other  hand,  twin 
structure  is  extremely  common,  the  twin  plane  being  most 
frequently  either  the  face  of  the  same  or  some  other  rhom- 
bohedron  or  the  basal  pinakoid,  the  latter  not  being  one  of 
their  planes  of  symmetry.  Fig.  272  is  a  common  twin 
group  of  Calcite,  in  which  two  rhombohedra  of  the  same 


FIG.  273. 


sign  are  twinned  upon  a  face  of  the  more  obtuse  rhombohe- 
dron  —  \  R.  The  faces  in  front  meet  in  a  re-entering  angle, 
and  those  below  in  a  parallel  salient  one,  the  twin  edges  in 
both  cases  being  parallel  to  the  longer  diagonals  of  the  faces, 
while  those  of  the  side  faces  are  parallel  to  the  middle  edges 
of  the  rhombohedron.  When  this  structure  is  repeated  by  the 
addition  of  a  third  crystal,  as  in  fig.  273,  the  middle  member 


CHAP.  IX.]       Rhombohedral  Twin  Crystals. 


173 


of  the  group  R'  is  often  reduced  to  a  thin  parallel  plate,  the 
third  one  R"  being  parallel  in  position  to  the  first  R;  and 
when  the  number  of  individuals  is  much  greater,  and  the 
intermediate  ones  are  very  thin,  the  group  is  scarcely  dis- 
tinguishable from  a  simple  crystal,  the  twin  structure  being 
only  apparent  in  the  numerous  fine  striations  covering  two 
of  the  faces  parallel  to  their  horizontal  diagonals,  and  the  other 
four  parallel  to  their  middle  edges.  Fig.  2  74  is  a  contact  twin 


FIG.  274. 


FIG.  275. 


of  the  hexagonal  prism  upon  a  face  of  the  same  rhombohedron 
—  jJ?,  and  fig.  275,  another  having  a  face  of  the  unit  rhom- 
bohedron as  a  twin  plane,  the  two  crystals  making  a  nearly 


FIG.  276. 


FIG.  277. 


right-angled  group,  the  inclination  of  the  principal  axes  to 
each  other  being  89°. 04'.  Fig.  276  is  a  contact  group  of  the 
common  scalenohedron  JR  3  of  Calcite  twinned  upon  a  face 
of  the  acute  rhombohedron  2  R. 


174 


Systematic  Mineralogy. 


[CHAP.  IX. 


In  the  second  case,  where  the  basal  pinakoid  is  the 
twin  plane,  the  axes  of  the  component  crystals  are  parallel. 
Fig.  277  is  the  contact  twin  or  hemitrope  of  a  single  rhom- 


FIG.  278. 


FIG.  279. 


bohedron,  and  fig.  278  the  same  completely  developed  as  a 
penetration  twin  of  two.    Fig.  279  is  a  hemitrope  of  the  com- 
mon scalenohedron  R  3  of  Calcite,  also  upon  the  basal  plane ; 
FIG.  280.  and   fig.  280  a  similar  twin  of  the 

combination  oo  P.  —  \  R  in  the  same 
mineral.  Here  the  two  prisms  are  in 
contact,  and  their  section  being  a 
regular  hexagon,  there  are  no  re- 
entering  angles,  but  the  compound 
character  is  apparent  from  the  shape 
of  the  prism  faces,  which  are  alter- 
nately rectangular  and  six-sided,  in- 
stead of  all  being  irregular  five-sided 
figures,  as  in  the  single  crystal,  fig.  144. 
Fig.  281  is  a  contact  twin  of  the  com- 
mon combination  en  P.R.—R  of  Quartz,  the  faces  of  the 
positive  rhombohedron  in  one  crystal  lying  parallel  with 


CHAP.  IX.]         Tetragonal  Twin  Crystals. 


175 


FIG.  281. 


those  of  the  negative  one  in  the  other.     This  is  a  case  of 
partial  penetration,  the  faces  adjacent  to  the  surface  of  con- 
tact, which  is  not  the  twin  plane,  meet- 
ing in  re-entering  angles;  but  when,  as 
very  frequently  happens,  there  is  more 
complete  penetration,  these  angles  are 
convex,  and  the  group   can    only  be 
distinguished  from  a  simple  crystal  by 
the  irregular  character  of   the   faces, 
which  rarely  have  even  surfaces,  por- 
tions   of    one     rhombohedron    being 
irregularly     distributed     through     the 
other  in  a  manner  which   shows    the 
crystals  to  be  combined,  and  that  their 
contact  is  jiot  in  a  plane  surface.     A  complete  description 
of  twin  structures  of  this  kind  will  be  found  in  Descloizeaux's 
memoir  on  Quartz. 

Twin  crystals  of  the  tetragonal  system.  In  the  type  of 
twin  structure  most  frequently  observed  in  this  system,  the 
twin  plane  is  a  face  of  the  pyramid  of  the  second  order, 


FIG.  282. 


'Fie.  283. 


m  Pec.  Fig.  282,  one  of  the  simplest  examples,  is  a  hemi- 
trope  of  the  tetragonal  pyramid  in  Hausmannite,  the  lower 
half  being  rotated  to  the  left  on  the  lower  plane  (oil).  This, 
when  repeated  symmetrically  upon  all  four  sides  of  the 
pyramid,  gives  the  group  of  five  individuals,  fig.  283. 


Systematic  Mineralogy. 


[CHAP.  IX. 


Another  very  common  example  of  the  same  kind  (fig.  284) 
occurs  in  the  combination  P.  GO  P  of  Tinstone,  and  this,  when 
repeated  with  a  third  individual,  the  middle  one  being 
shortened  to  a  parallel  plate,  gives  the  bent-kneed  or  geni- 

FIG.  284.  FIG.  285. 


culated  group  (fig.  285)  whose  ends  are  both  in  the  direct 
position,  the  middle  one  alone  being  reversed.  Fig.  286  is 
another  example  of  a  triple  group,  very  characteristic  of  the 
allied  species  Rutile.  Here  the  ends  are  bent  away  from  the 
middle,  the  twin  planes  being  different  faces  of  Pec. 


FIG. 


FIG.  287. 


In  sphenoidal  hemihedral  forms,  the  twin  plane  is  commonly 
a  face  of  a  pyramid,  and  when  the  proportions  of  the  latter 
differ  but  slightly  from  those  of  a  regular  octahedron,  the 
twin  groups,  whether  contact  or  penetration,  are  very  like 
tetrahedral  twins  in  the  cubic .  system.  This  is  especially 


CHAP.  IX.] 


Rhombic  Twin  Crystals. 


177 


observed  in  Copper  Pyrites.  In  the  pyramidal  hemihe- 
dral  forms  the  twin  plane  is  usually  a  face  of  the  diagonal 
prism  QO  P  GO,  or  the  twin  axis  is  one  of  the  lateral  axes.  This 
in  crystals  like  fig.  287  has  the  effect  of  bringing  the  faces  SS 
of  the  pyramids  of  the  third  order  ^(3^3)  into  a  re-entering 
or  negative  solid  angle  in  the  basal  section  of  the  pyramid 
of  the  second  order  P<x>,  but  when  these  planes  are  less 
completely  developed,  they  usually  appear  as  contrasted 
diagonal  striations  upon  the  faces  of  Pec. 

Twin  crystals  of  the  rhombic  system.  Crystals  twinned 
upon  the  faces  of  pyramids  or  prisms  are  of  frequent  occur- 
rence in  this  system.  Some  of  the  most  familiar  examples 


FIG.  2 


FIG.  289. 


are  afforded  by  the  allied  species,  Aragonite  and  White-lead 
Ore.  Fig.  288  is  a  contact  twin  of  Aragonite  on  the  face 
(i  i  o)  of  the  prism  oo  />,  and  fig.  289  its  section  on  the  basal 
plane,  the  shading  lines  being  parallel  to  the  brachypinakoid 
in  each  individual,  which  faces  meet  on  one  side  in  a  pro- 
jecting, and  on  the  other  in  a  re-entering,  angle.  If,  as  very 
commonly  happens,  the  hollow  space  a  is  filled  up  by  parallel 
elongation  of  the  two  crystals,  the  group  may  resemble  a 
single  crystal  if  the  development  is  confined  to  the  zone  of 
the  prism.  Figs.  2900,  290^  give  the  same  group  with  a  third 
individual,  the  twinning  being  repeated  on  the  same  face ; 
this  brings  the  prism  faces  in  i.  and  in.  into  similar  position, 
the  middle  crystal  n.  being  reduced  to  a  parallel  plate. 

N 


ij 8  Systematic  Mineralogy.  [CHAP.  ix. 

This  is  also  a  very  common  case,  the  middle  individual 
being  many  times  repeated,  and  appearing  as  a  series  of  fine 

FIG.  2900.  FIG.  290*5. 


striations  parallel  to  the  twin  plane.  Fig.  291  is  another 
group,  in  which  the  ends  i.  and  in.  are  twinned  upon  adjoining 
faces  of  the  prism  in  the  middle  crystal  n.  Fig.  292  is  the 


FIG.  291. 


FIG.  292. 


horizontal  section  of  a  similar  triple  group  of  Copper-glance, 
twinned  upon  faces  of  oo  P.  The  angle  of  the  prism  in  this 
species  is  so  nearly  =  120°  (119°  35')  that  the  group  is  very 
similar  in  appearance  to  a  regular  hexagonal  prism. 

Figs.  293,  294  are  examples  of  cruciform  twins  produced 
by  the  penetration  of  two  crystals  of  the  combination 
oo  P.  oo  Pao.oPm  Staurolite.  In  fig.  293  the  twin  plane  is 
a  face  (032)  of  f  Px>,  the  vertical  arms  of  the  cross  being 
in  direct,  and  the  transverse  ones  in  the  inverted  position. 


CHAP.  IX.]  Rhombic  Twin  Crystals. 


179 


The  latter  are  nearly,  but  not  quite,  horizontal,  the  angles  be- 
tween the  vertical  axes  of  the  two  crystals  being  alternately 

FIG.  293.  FIG.  294 


91°  36'  and  88°  24'.  In  fig.  294  the  twin  plane  is  the  face 
(232)  of  I-/5!,  the  left-hand  crystal  being  placed  in  direct 
and  the  right-hand  one  in  reversed  position.  Here  the  axes 
of  the  two  prisms  cross  at  58°  46',  and  the  re-entering  angle 
between  the  brachypinakoid  faces  is  119°  34',  or  nearly 
120°.  These  peculiarities  are  caused  by  the  fundamental 
ratios  a  :  b  :  c  of  this  species  being  very  nearly  as  \  :  i  :  f ; 
the  actual  values  are  0.48  :  i  :  0.67,  which  give  a  twin 
axis  inclined  to  the  vertical  at  nearly  45°  in  the  first  case, 
and  60°  in  the  second. 

In  the  sphenoidal  heraihedral  forms  of  this  system,  the 
three  pinakoids  are  possible  twin  planes ;  FIG.  295. 

but  examples  of  such  twinning  are  not 
common.  The  best  known  case  oc- 
curs in  Manganite  where  a  combination 
containing  f-  J^QO  or  {365}  as  a  sphenoid 
is  twinned  upon  the  brachypinakoid. 

Fig.  295  is  a  compound  structure 
observed  in  the  hemimorphic  species, 
Electric  Calamine;  the  twin  plane  is 
the  basal  pinakoid,  whose  normal  is  the  vertical  or  hemi- 
morphic  axis.     As,  however,  the  faces  of  2  P  2    in  these 

N   2 


i8o 


Systematic  Mineralogy.  [CHAP.  IX. 


crystals  only  appear  at  the  negative  end  of  the  vertical  axis, 
it  is  necessary  to  consider  the  lower  component  as  inverted 
about  one  of  the  lateral  axes  as  well  as  about  the  twin  axis, 
and  the  notation  of  the  faces  of  the  inverted  form  will  differ 
according  as  this  reversal  takes  place  about  a  or  b.  In  the 
first  case  the  front  transverse  faces  3  P<*>,  &>P,  2  p2  will  all 
be  positive  to  the  axis  a,  while  the  longitudinal  or  side  faces, 
oo  P<x>  and  P<x>,  on  the  same  side  will  be  positive  above  and 
negative  below  to  the  axis  b  ;  and  in  the  second  the  longi- 
tudinal faces  on  the  same  side  will  have  similar  signs,  while 
the  transverse  ones  will  be  negative  to  the  axis  a  in  the  lower 
crystal.  This  difference  is  only  geometrical,  and  there  is  no 
reason  to  consider  either  reading  as  preferable  to  the  other. 
The  same  structure  may  also  be  explained  by  supposing  the 
vertically  reversed  crystals  as  essentially  penetrating  the 
direct  one,  which,  however,  requires  either  the  brachy- 
pinakoid  or  macropinakoid  to  be  the  twin  face,  an  assump- 
tion which  is  not  compatible  with  the  exclusion  of  planes  of 
symmetry  from  possible  twin  planes. 

Twin  crystals  of  the  oblique  system.     In  this  system  any 
face  is  a  possible  twin  plane  except  the  clinopinakoid,  or 
FIG.  296.         plane  of  symmetry,  and  of  the  three  crystal- 
lographic  axes  only  the  vertical  is  a  possible 
twin  axis.     The  type  of  most  frequent  occur- 
rence is  that  having  the  orthopinakoid  as  the 
twin  plane,  the  twin  axis  being  horizontal. 
When  the  same  face  is  the  plane  of  contact, 
the  group  resembles  fig.  296,  a  hemitrope  of 
the  combination  —  P.  ao  P.  00  jPoo,  common  in 
Gypsum,  the  front  half  of  the  crystal  being  sup- 
posed to  be  the  reversed  one.  The  faces  of  the 
hemipyramids  in  the  two  crystals  meet  in  re- 
entering  angles  above  and  parallel  salient  ones 
below,  and  their  clinopinakoids  in  the  same  surface  at  either 
end.   When  the  individuals  are  differently  proportioned,  and 
have  their  greatest  length  parallel  to  the  inclined  axis,  they 


CHAP.  IX.] 


Oblique  Twin  Crystals. 


181 


may  form  a  complete  cruciform  penetration  twin,  the  re- 
entering  angles  of  the  faces  of  the  hemipyramid  breaking 
the  lines  of  the  vertical  edges  in  front  and  behind.  These 
may  be  also  considered  as  twinned  by  rotation  upon  the  ver- 
tical axis,  in  which  case  the  twin  plane  is  horizontal  and  not 
a  possible  crystallographic  face ;  while  on  the  former  view 
the  normal  to  the  twin  plane,  being  horizontal,  is  not  a  pos- 
sible crystallographic  axis. 

The  same  type  of  twin  crystal  is  very  common  in  the 

FIG.  297.  FIG.  29 


species   Orthoclase,   two    individuals   of  the  combination 
co  P,  oP,  2  fee,  GO  ^oo  (fig.  297),  being  combined  with 


partial    penetration   upon  the   clinopina-  FIG.  299. 

koid  in  the  groups,  figs.   298,  299,  each 

component    preserving    its    individuality, 

except  in  the  case  of  the    prism    faces, 

which,  though  apparently  simple,  are  made 

up  of  parts   of  different   crystals,  joined 

along  the  diagonal  dotted  lines   in   the 

figures.     This  is  known  as  the  Carlsbad 

type   of  twin    crystal,  the    groups    being 

further  distinguished   as  right-   and  left- 

handed,  according  as  the  reversed  crystal 

is  to  the  right  (fig.  298)  or  left  (fig.  299) 

of  the  direct  one,  a  geometrical  distinction  which  is  only 


1 82 


Systematic  Mineralogy.  [CHAP.  IX. 


recognisable  so  long  as  the  penetration  is  incomplete. 
Fig.  298  is  noted  as  having  a  horizontal  twin  axis,  and 
fig.  299  a  vertical  one. 

Fig.  300  is  a  similar  combination  to  fig.  296,  the  positive 
hemi-orthodome  P<x>  being  substituted  for  the  more  acute  one 


FIG.  300. 


FIG.  301. 


This,  when  twinned  by  rotation  upon  a  plane  parallel 
to  the  base,  as  shown  in  the  dotted  line,  gives  fig.  301,  where 
the  faces  of  the  prism  meet  in  re-entering  angles  in  front 
and  projecting  ones  behind.  The  twin  axis  in  this  case  is  in- 
clined to  the  vertical  at  the  angle  180°  —  (90  +  /3°).  This 
is  sometimes  called  the  Manebach  type  of  twin  in  Felspar, 

Fig.  302  is  a  crystal  of  the  same  combination  as  the  last, 
but  of  a  nearly  square  section  on  the  orthopinakoid  ;  this, 


FIG.  302. 


FIG.  303. 


when  twinned  on  the  clinodome  2  ;Pao,  indicated  by  the 
dotted  line,  produces  fig.  303,  in  which  there  are  no  re- 
entering  angles.  This  is  known  as  the  Baveno  type  of  twin 
in  Felspar. 


CHAP.  IX.]  Triclinic  Twin  Crystals. 


183 


Twin  crystals  of  the  triclinic  system.  As  there  are  no 
planes  of  symmetry  in  this  system,  any  face  is  a  possible  twin 
plane.  The  observed  cases  are  generally  similar  to  those  of  the 
oblique  system,  with  an  additional  one  special  to  the  system 
where  the  brachypinakoid  is  the  twin  plane.  This  is  repre- 
sented in  fig.  304,  the  right-hand  half  of  the  divided  crystal 
being  supposed  to  be  rotated  upon  a  normal  to  the  brachy- 
pinakoid, which  brings  the  opposite  halves  of  the  basal 
pinakoids  together  in  a  re-entering  angle  at  the  top,  as  indi- 


FIG.  304. 


FIG.  305. 


cated  by  the  arrows,  and  in  a  corresponding  salient  one  at 
the  bottom ;  and  those  of  the  hemi-macrodome  in  the  same 
way  are  convex  in  front  and  concave  behind.  Supposing 
such  a  group  to  have  only  its  lower  faces  developed,  it  would 
be  impossible  to  distinguish  it  from  a  simple  crystal  by  con- 
siderations of  form  alone.  If  this  structure  is  repeated  with 
a  third  individual,  as  in  fig.  305,  the  exterior  components 
resemble  the  halves  of  crystals  in  their  normal  positions, 
divided  by  an  intermediate  parallel  plate.  This  middle  in- 
dividual may,  however,  be  replaced  by  a  large  number  of 
much  thinner  plates,  in  which  case  the  re-entering  angles  of 
the  basal  pinakoids  appear  as  fine  striations  parallel  to  the 
edge  between  GO  ^Poo  and  o  P  upon  the  latter  faces.  These, 
known  as  polysynthetic  striations,  are  especially  characteristic 
of  the  Felspars  crystallising  in  this  system,  there  being  no  ap- 
parent limit  to  the  number  and  fineness  of  such  twin  lamellae, 
which  are,  as  a  rule,  easily  recognised  by  the  microscope, 


1 84 


Systematic  Mineralogy. 


[CHAP.  IX. 


even  when  not  apparent  to  the  naked  eye.     In  many  cases, 

however,  they  are  perfectly  visible  without  being  magnified, 

as,  for  instance,  in  Labradorite. 

Fig.  306  is  an  example  of  complex  twin  grouping  of 

triclinic  crystals,  the  individuals  i.  n.  and  in.  iv.  being 
FIG.  306.  twinned  upon  the  brachypinakoid,  as  in  the 
preceding  example,  and  these  are  further 
compounded  according  to  the  Carlsbad  type. 
This,  though  generally  similar  in  appearance 
to  fig.  305,  differs  from  it  by  having  similar 
faces  arranged  in  pairs  instead  of  alternating 
singly.  The  above  are  the  principal  types 

;/  of  twin  crystals  in  the  different  systems. 
Other  and  more  special  cases  will  be  noticed 
in  treating  of  the  minerals  in  detail. 
Irregularly  developed  crystals.  In  the  artificial  prepara- 
tion of  crystals  in  the  laboratory,  it  is  a  well-known  practice 
to  select  some  of  the  most  regular  individuals  from  those 
first  deposited,  and  place  them  in  such  a  manner  in  fresh 
portions  of  the  solution  that  they  may  increase  as  equally  as 
possible  in  all  directions,  for  which  purpose  it  is  necessary 
to  alter  their  positions  from  time  to  time.  If,  on  the  other 
hand,  the  growth  upon  any  part  is  hindered,  as,  for  example, 
upon  the  face  in  contact  with  the  surface  of  the  vessel,  the 
deposition  of  the  additional  material  will  increase  the  faces 
remaining  free  to  such  an  extent  that  their  true  character 
may  not  be  readily  seen.  This  condi- 
tion prevails  to  a  great  extent  in  natural 
crystals,  and  in  fact  one  of  the  chief 
objects  of  determinative  crystallography 
is  the  reduction  of  such  distorted  forms 
to  their  theoretical  regularity.  Only  a 
few  of  the  simpler  cases  can  be  given 
in  this  place.  Fig.  296  is  an  octa- 
hedron with  two  faces  (111)  (in)  prominently  larger 
than  the  other  six,  in  which  the  solid  angles  are  no  longer 


FIG.  307. 


CHAP.  IX.] 


Distorted  Crystals. 


i8S 


apparent,  as  the  four  faces  similarly  placed  with  respect 
to  any  principal  axis  do  not  meet  in  the  same  point,  but  in 
an  edge  parallel  to  the  axis  of  a  dodecahedral  zone,  or,  in 
other  words,  the  crystal  is  elongated  in  the  zones  [  1 1  o  ], 
[011],  and  [  i  o  i  ],  and  shortened  on  [  1 1  o  ],  or  shortened 
on  the  ternary  axis  (i  1 1).  This  is  common  in  crystals  of 
Nitrate  of  Barium  deposited  from  solution  upon  a  flat 


FIG.  308. 


FIG.  309. 


surface,  the  enlarged  faces  appearing  as  nearly  regular  hexa- 
gons ;  and  it  is  also  readily  obtained  by  cleavage  from  octa- 
hedra  of  Fluorspar.  When  the  octahedron  is  elongated  on  a 
ternary  axis  the  faces  perpendicular  to  that  axis  may  be 
completely  obliterated,  producing  an  acute  rhombohedron, 
as  in  fig.  308.  This  is  easily  obtained  by  cleavage  in  Fluor- 
spar, and  its  octahedral  character  is  as  easily  restored  by 
cleaving  off  two  tetrahedra  in  the  directions  shown  in  the 
dotted  lines.  Fig.  308,  an  octahedron  distorted  by  elonga- 
tion on  a  binary  axis,  has  a  general  re-  FIG.  310. 
semblance  to  a  combination  of  two 
prismatic  forms  in  the  rhombic  system. 

Fig.  311  is  a  rhombic  dodecahedron 
elongated  vertically,  which  converts  the 
upright  faces  into  a  square  prism,  and 
the  inclined  ones  being  unchanged,  the 
general  effect  is  that  of  the  combination 
of  a  prism  and  pyramid  of  diagonal 
position  co  P.  P  co  in  the  tetragonal  system. 

In  the  hexagonal  system  distorted  crystals  are  also  of  fre- 


1 86 


Systematic  Mincralog}>.  [CHAP.  ix. 


quent  occurrence,  producing  great  variation  in  shape.  Some 
of  the  most  familiar  examples  are  afforded  by  the  common 
combination  ft  —  J?  <x>  R  in  Quartz,  three  of  which  are  repre- 
sented in  the  following  figures.  In  fig.  311  the  four  rhom- 
bohedral  and  two  prismatic  faces  in  the  zone  [2112]  are 

FIG.  311.  FIG.  312. 


lengthened  parallel  to  its  axis  ;  in  fig.  312  the  crystal  is 
elongated  on  the  axes  of  the  zones  [i2?o],  [1212],  and 
[1212],  giving  an  apparent  rhombic  character  to  the  com- 
FIG.  313.      bination  ;  and  in  fig.  3 13  one  face  of  one  rhom- 
bohedron  is  prominently  longer  than  the  others, 
giving    a    kind    of   chisel-edged     termination. 
Crystals  of  this  kind  generally  with  only  one 
end  developed  are  common  among  the  brilliant 
groups  of  rock  crystal    found  in  the  Western 
Alps  of  Dauphine  and  Piedmont 

In  the  rhombic  and  oblique  systems  the 
commonest  case  of  distortion  is  that  of  the  un- 
equal development  of  the  faces  of  the  prism,  one  pair  being 
broader  than  the  other,  giving  a  rhomboidal  instead  of  a 
rhombic  basal  section. 

Imperfections  in  the  faces  of  crystals.  In  addition  to  the 
irregularities  arising  from  distortion  and  compound  structure, 
crystals,  when  of  large  size,  often  appear  with  roughened, 
striated,  or  even  partially  hollow  faces.  The  latter  im- 
perfection is  common  in  substances  that  crystallise  easily, 
whether  from  solution,  as  salt  and  alum,  from  sublimed 
vapours  as  arsenious  acid,  or  from  molten  masses,  as  lead, 


CHAP.  IX.]  Imperfections  of  Crystals.  187 

silver,  bismuth,  and  silicates  produced  in  furnace  slags.  In 
all  these  cases  crystals  are  often  observed  having  their  edges 
perfectly  denned,  while  the  faces  themselves  are  hollowed 
out  and  reduced  to  very  narrow  surfaces  adjacent  to  the 
edges,  giving  skeleton  structures,  in  which  the  general  ele- 
ments of  the  form  are,  however,  usually  recognisable  without 
difficulty.  In  Fluorspar,  octahedra  with  roughened  faces  are 
occasionally  found,  which  are  made  up  of  minute  cubes  piled 
up  like  courses  of  masonry,  the  side  of  each  successive  course 
being  diminished  by  the  breadth  of  two  cubes.  The  roughness 
of  the  face  is,  therefore,  due  to  the  step-shaped  section  of  the 
pile. 

In  the  hexagonal  system,  instances  of  irregular  single 
crystals,  built  up  from  smaller  individuals  of  the  same  or 
different  kinds,  are  very  common  in  Calcite.  In  the  north 
of  England  lead  mines  the  obtuse  rhombohedron  known  as- 
Nailhead  Spar,  and  the  combination  shown  in  fig.  144,  are 
often  aggregated  in  such  a  manner  as  to  produce  rough- 
faced  rhombohedra  and  scalenohedra  often  of  considerable 
size.  Sometimes  this  step-shaped  outline  is  apparent  on 
some  of  the  faces,  while  the  others  are  comparatively  smooth 
and  regular.  The  same  kind  of  structure  may  often  be 
brought  out  in  the  most  regularly  developed  crystals  by  the 
action  of  solvents,  which  produce  the  so-called  corrosion 
figures,  which  show  that  in  many  cases  the  smallest  crystals- 
are  fully  as  complex,  or  even  more  so,  than  those  of  larger 
size.  The  characteristic  habit  of  particular  crystals  is  often 
apparent  in  the  most  minute  individuals  ;  for  instance,  the 
peculiar  geniculated  groups  of  Tinstone  and  Rutile  (fig. 
284),  are  developed  when  these  oxides  are  crystallised 
from  solution  in  melted  borax  or  phosphate  of  soda  before 
the  blowpipe,  even  when  the  crystals  must  be  magnified  from 
400  to  500  diameters  to  render  them  visible. 

The  prismatic  faces  of  Quartz  crystals  are  very  generally 
covered  with  horizontal  striations,  representing  very  minute 
portions  of  the  faces  of  an  acute  rhombohedron.  This  is 


1 88  Systematic  Mineralogy.  [CHAP.  IX. 

generally  called  oscillatory  combination,  a  tendency  towards 
the  formation  of  rhombohedral  ends  being  supposed  to  have 
alternated  with  another  towards  prismatic  elongation.  In 
Beryl  the  prisms  are  striated  vertically,  as  is  also  the  case  in 
Tourmaline,  the  continued  repetition  of  the  prisms  of  the  first 
and  second  order  producing  nearly  cylindrical  forms. 

Another  class  of  imperfection,  where  the  faces  of  crystals 
are  curved  instead  of  plane  surfaces,  is  characteristic  of 
certain  minerals,  the  most  striking  examples  being  afforded 
by  Siderite,  which  occurs  in  rhombohedra  having  strongly 
curved  faces  ;  Gypsum  and  Diamond  :  the  latter,  when 
in  the  form  of  the  hexakisoctahedron,  are  often  nearly 
spherical  in  shape. 

Habit  of  crystals.  In  describing  minerals  it  is  usual  to 
speak  of  their  crystals  as  affecting  particular  types,  according 
to  the  character  of  the  dominant  or  principal  faces  ;  thus, 
in  the  cubical  system,  they  may  be  cubic,  octahedral,  dode- 
cahedral,  &c,  as  one  or  other  of  the  principal  forms  prevail 
in  the  combination.  In  the  other  systems  the  types  are 
pyramidal,  sphenoidal,  rhombohedral,  or  scalenohedral  when 
the  closed  forms  are  most  apparent,  and  prismatic  when  the 
development  is  mainly  in  the  direction  of  the  open  forms. 
In  the  latter  case  several  further  distinctions  are  founded 
upon  the  relation  of  the  height  of  the  prism  to  the  breadth 
of  its  base  or  closing  pinakoid.  The  shorter  forms,  or  those 
having  their  principal  dimensions  in  the  direction  of  the  lateral 
axis,  are  said  to  be  tabular,  or,  if  very  thin,  platy  or  scaly  : 
when  the  height  is  only  a  few  times  the  breadth,  they  are  short 
columnar,  and  as  the  relative  length  of  the  prismatic  axis 
increases  they  become  columnar,  slender-prismatic,  and 
acicular,  or  needle-shaped.  In  these  terms  the  expression 
prismatic  is  not  restricted  to  the  forms  assumed  as  vertical 
prisms,  but  is  used  with  the  general  signification  of  any  zone 
of  prismatic  planes,  whether  vertical  or  horizontal.  The 
habit  of  considering  any  prominently  defined  axis  as  a 
prismatic  one  in  describing  the  appearance  of  crystals  is  very 


CHAP.  IX.]  Irregular  Aggregates.  1 89 

general  and  convenient,  but  care  must  be  taken  not  to  con- 
found such  '  columnar '  forms  with  the  more  exactly  deter- 
mined crystallographic  prisms. 

Irregular  grouping  of  crystals.  Masses  of  crystals,  when 
not  arranged  as  symmetrically  twinned  forms,  are  spoken  of 
as  groups  or  crystalline  aggregates.  These  are  commonly 
found  in  hollow  spaces  or  druses  in  the  containing  rock, 
attached  at  one  end,  with  the  faces  terminating  the  opposite 
end,  freely  developed,  the  individuals  of  the  group  having  a 
more  or  less  radial  arrangement  diverging  from  the  point  of 
attachment.  This,  in  general  terms,  may  be  considered  as  the 
most  typical  kind  of  grouping  of  well  individualised  crystals. 
When  the  aggregates  are  of  a  more  compact  kind,  the  indi- 
viduals are  rarely  recognisable  with  anything  like  their  full 
number  effaces,  but  appear,  as  a  rule,  as  columnar  or  fibrous 
masses  arranged  in  parallel  or  divergent  forms.  The  latter, 
when  in  sufficient  numbers,  make  up  more  or  less  spheroidal 
masses,  which,  according  to  the  size  of  the  spheroids,  are 
spoken  of  as  mamillary,  remform,  or  kidney- shaped,  and 
botryotdal,  or  grape-like  masses  or  concretions.  Other  smooth 
spheroidal  masses  of  substances,  having  no  apparent  definite 
structure,  are  generally  called  nodules. 

Parallel  aggregates  of  a  fibrous  structure,  such  as  those 
of  Calcite  and  Gypsum,  often  form  regular  beds  of  a  silky 
character  on  the  face.  These  are  known  as  Satin  Spar ;  the 
same  structure  is  common  in  salt  and  alum,  where  the  fibres 
are  often  bent  or  contorted.  Aggregates  resembling  corals, 
mosses,  and  other  organised  forms,  are  common  in  Arra- 
gonite,  the  so-called  flos-ferri,  or  flowers  of  iron,  and  native 
metals ;  the  latter  are  usually  called  dendritic  forms,  the 
same  name  being  also  applied  to  the  plant-like  stains  of 
Brown  Iron  Ore  and  Peroxide  of  Manganese  on  rocks.  Wire- 
like  or  filiform  masses  are  very  characteristic  of  native  silver. 

Stalactites  are  irregularly  shaped  crystalline  masses  found 
in  caverns  hanging  from  the  roof,  and  stalagmites  are 
similar  masses  accumulated  above  the  floor.  These  terms 


190  Systematic  Mineralogy.  [CHAP.  X, 

are  rather  geological  (as  indicating  methods  of  origin)  than 
structural. 

When  no  indications  of  crystalline  structure  are  apparent 
in  a  mineral  aggregate  it  is  said  to  be  massive.  Sometimes 
such  masses  are  spoken  of  as  amorphous.  This,  however,  is 
an  improper  use  of  the  term,  which  should  only  be  applied 
to  substances  which  fail  to  show  crystalline  structure  when 
tested  by  optical  and  other  methods,  many  perfectly  well 
crystallised  bodies  often  appearing  structureless  until  so 
examined. 


CHAPTER  X. 

MEASUREMENT   AND   REPRESENTATION   OF   CRYSTALS. 

THE  angles  between  two  faces  in  a  crystal  may  be  measured 
in  two  different  ways,  namely,  directly  from  the  inclination 
of  a  pair  of  jointed  blades  striding  over  the  edge,  and  in- 
directly by  determining  the  angle  through  which  the  crystal 
must  be  turned  in  order  to  obtain  the  reflected  image  of  an 
object  successively  from  both  faces.  The  instrument  used  in 
the  first  method  is  the  hand  or  contact  goniometer,  which 
has  not  been  materially  altered  from  the  form  in  which  it 
was  originally  made  by  Carangeot  for  Rome  de  ITsle  and 
Haiiy  at  the  end  of  the  last  century,  and  is  shown  in  fig.  314. 
It  consists  of  a  semicircular  arc,  divided  into  single  or  half 
degrees,  according  to  size,  having  attached  to  it  two  steel 
blades,  one  of  which,  k  m,  is  fixed,  or  rather  has  only  a 
sliding  movement  in  a  straight  line  upon  the  pins  c  d,  while 
the  other  has  an  angular  as  well  as  a  sliding  motion  about  c. 
The  zero  point  of  the  graduation  is  on  a  line  parallel  to  the 
direction  of  the  fixed  rule,  so  that  when  the  latter  is  laid 
upon  one  of  the  faces  containing  the  angle  to  be  measured, 
and  the  movable  one  turned  until  it  bears  similarly  upon  the 
other  face,  the  edge  g  i  will  indicate  the  value  of  the  angle 


CHAP.  X.] 


Contact  Goniometer, 


191 


upon  the  divided  arc.  The  object  of  the  slots  is  to  allow 
the  length  of  the  measuring  arms  to  be  varied  to  suit  the 
size  of  the  crystal.  For  the  same  purpose  the  arc  is  divided, 
and  can  be  folded  back  upon  a  hinge  at  b,  the  supporting 


arm/  being  attached  by  a  screw  at  the  back,  which  can  be 
taken  out  when  greater  freedom  of  manipulation  is  required, 
as,  for  instance,  when  working  upon  crystals  in  implanted 
groups.  A  more  convenient  arrangement  is  to  have  the 
arms  separate  from  the  measuring  arc,  in  which  case  a  com- 
plete circle  may  be  advantageously  used. 

Under  the  most  favourable  conditions  angles  can  be 
measured  by  the  contact  goniometer  to  within  half  a  degree, 
when  the  crystals  are  of  a  certain  size.  It  is  therefore  best 
adapted  for  trial  measurements  upon  large  and  imperfectly 
lustrous  faces,  or  for  use  in  making  models.  When  greater 
precision  is  required,  the  more  exact  method  depending 
upon  reflection  from  the  faces  must  be  used.  Fig.  315 
represents  Wollaston's  reflecting  goniometer  in  one  of  its 
simplest  forms.  It  consists  of  a  circular  disc  E  E,  with  a 
divided  rim  reading  to  single  minutes  of  arc  by  a  vernier  at 
R,  movable  about  a  horizontal  axis  passing  through  the 
bearing  c  at  the  top  of  the  pillar  B,  with  a  milled  head  for 
turning  it  at  G.  This  axis  is  hollow,  and  an  inner  one,  moved 


192 


Systematic  Mineralogy, 


[CHAP.  X. 


by  the  milled  head  i,  is  connected  with  the  arrangement 
carrying  the  crystal  at  K.  This  consists  of  two  bent  arms  K  M, 
movable  about  a  pin  L,  and  the  carrier  proper  o,  which  has 
a  rotatory  as  well  as  a  sliding  movement  in  the  collar  N, 


the  crystal  a  being  attached  by  a  ball  of  wax  to  its  outer  end. 
The  parts  s  T  and  u  form  a  clamp  upon  the  disc  F  for  main- 
taining the  circle  in  any  particular  position.  The  foot  A  is 
a  block  of  wood  ;  in  the  larger  instruments  it  is  of  metal, 
with  levelling  screws. 

In  observing  angles  it  is  necessary  to  bring  the  edge 
exactly  into  the  line  of  the  axis  of  the  instrument,  which  is 
done  by  the  various  movements  of  the  carrier,  which  together 
form  a  kind  of  universal  joint.  Two  signals  are  required, 
one  of  which,  x,  is  seen  by  reflection,  and  the  other,  y,  di- 
rectly. These  may  be  any  small,  well-defined  objects,  such 
as  a  bright  spot  seen  through  a  screen  placed  before  a  win- 
dow or  a  lighted  lamp,  and  a  window-bar,  chalk-line  or  line 


CHAP.  X.]  Reflecting  Goniometer.  193 

of  light,  as  far  off  as  they  may  be  conveniently  seen.  The 
edge  is  in  adjustment  when  the  image  of  a  horizontal  or 
vertical  line  reflected  from  either  face  coincides  with  that 
viewed  directly  from  the  same  point,  the  eye  being  as  close 
to  the  crystal  as  possible.  The  circle  being  clamped,  the 
crystal  is  then  turned  by  i  until  the  reflected  image  of  x 
is  brought  into  coincidence  with  y,  seen  directly,  when  it  is 
undamped,  and  the  whole  circle  is  turned  by  G  until  the 
same  thing  is  seen  from  the  second  face.  The  angle  through 
which  the  circle  is  rotated  will  be  the  supplement  of  the 
dihedral  angle  required,  if  it  was  originally  set  to  zero.  As  a 
rule,  however,  it  is  better  not  to  start  from  this  point,  but  to 
read  the  vernier  after  each  adjustment,  and  take  the  differ- 
ence of  the  readings,  repeating  them  for  several  different 
positions  on  the  circle  to  eliminate  the  errors  of  eccentricity 
as  much  as  possible.  In  all  cases  the  observation  must  be 
often  repeated  to  obtain  results  of  any  value. 

Numerous  modifications  and  improvements  of  the  re- 
flecting goniometer  have  been  introduced  for  the  purpose  of 
facilitating  the  operation  of  centering  and  adjusting  the 
crystal,  and  obtaining  greater  precision  in  observation.  For 
the  latter  purpose  a  sighting  telescope  of  low  magnifying 
power  is  used,  the  observed  object  being  the  image  of  the 
cross  wires  in  a  second  telescope,  or  collimator ;  and  for  the 
former  the  carrier  is  made  with  a  system  of  rectilinear  and 
cylindrical,  or  sliding  and  rocking,  movements  analogous 
to  those  of  the  '  universal '  machine  tools.  The  more  exact 
modern  instruments  of  this  kind  are  usually  constructed  upon 
the  pattern  introduced  by  Mitscherlich  in  1843  ;  another 
construction,  that  of  Babinet,  in  which  the  divided  circle  is 
horizontal,  being  also  considerably  used.  The  latter  has 
the  advantage  of  placing  the  crystal  upright,  and  not  as  in 
the  preceding  form  overhanging  the  carrier ;  but  as  it 
requires  signals  that  are  not  in  the  same  vertical  plane,  it 
is  not  so  easily  used  when  without  a  telescope.  The  large 
instruments  made  by  Fuess,  of  Berlin,  on  this  pattern  are 

o 


194  Systematic  Mineralogy.  [CHAP.  X. 

remarkable  for  the  extreme  precision  of  their  centring  appa- 
ratus, which  has  two  straight  line  movements  at  right  angles 
to  each  other,  and  two  cylindrical  ones  also  at  right  angles, 
the  former  being  used  for  centring  or  bringing  the  edge  into 
view,  and  the  latter  for  adjusting  or  making  it  vertical.  This 
operation  is  facilitated  by  the  addition  of  a  lens  of  short 
focus  to  the  objective  end  of  the  telescope,  which  for  the 
time  converts  it  into  a  microscope  of  low  magnifying  power, 
and  allows  the  edge  to  be  brought  into  coincidence  with 
a  vertical  wire  in  the  eyepiece. 

When  the  observations  are  made  with  the  unaided  eye 
or  a  single  telescope,  the  signals  observed  must  not  be  too 
near,  or  there  may  be  an  appreciable  error  of  parallax.  A 
convenient  object  is  a  small  gas-flame  about  half-an-inch 
high,  from  15  to  30  feet  distant,  the  observations  being 
made  in  a  darkened  room.  When  the  instrument  has  two 
telescopes  the  error  of  parallax  may  be  eliminated,  but  the 
loss  of  light  is  so  considerable  that  crystals  with  very  perfect 
faces  are  required  to  give  distinct  images  of  the  signal. 

In  all  cases  observations  must  be  repeated  several  times, 
and,  if  possible,  upon  different  crystals,  the  quality  of  the 
reflection  obtained  being  specially  noted.  As  a  rule,  the 
smallest  crystals  have  the  most  brilliant  faces,  and  therefore 
give  the  best  results. 

A  goniometer  for  measuring  the  angles  of  minute  crystals 
with  dull  faces  has  recently  been  described  by  Hirschwald. 
It  is  of  the  Wollaston  pattern,  but  the  position  of  the  faces 
is  ascertained  by  bringing  them  into  the  focus  of  a  com- 
pound microscope  with  an  objective  of  very  short  focal 
distance,  which,  when  adjusted,  can  be  traversed  horizontally 
by  sliding  movements,  the  adjustment  of  the  face  being 
complete  when  its  whole  surface  is  equally  well  denned  in 
the  field  of  the  microscope  as  the  latter  is  moved  over  it 

The  calculation  of  the  crystallographic  constants  from 
the  observed  angles  of  crystals  is  the  work  of  determinative 
crystallography,  and  the  subject  is  beyond  the  scope  of  the 


CHAP.  X.]  Drawing  Crystals.  195 

present  treatise.  The  student  is  referred  to  the  special 
works  on  the  subject,  more  particularly  to  Miller's  tract  on 
Crystallography,  and  the  works  of  Klein  and  Mallard. 

Methods  of  representing  crystals.  The  larger  number  of 
figures  in  this  volume,  following  the  usual  practice  of  works 
on  Mineralogy,  are  drawn  in  so-called  parallel  perspective, 
which  supposes  the  point  of  sight  to  be  at  an  infinite  distance 
from  the  object  represented,  so  that  all  rays  proceeding  from 
the  latter  to  the  eye  are  parallel,  or  all  lines  and  surfaces 
parallel  to  each  other  remain  so  in  the  drawing ;  and  in  order 
to  bring  in  a  sufficient  number  of  faces  the  plane  of  projec- 
tion is  so  chosen  that  all  lines  actually  at  right  angles  will 
be  projected  as  acute  or  obtuse  angles,  and  consequently  all 
lines,  except  those  parallel  to  the  plane  of  projection,  and  all 
faces,  will  be  shown  in  other  than  their  true  dimensions. 
This  is  done  by  adopting  reduced  lengths  for  the  axes  in 
the  triaxial  systems,  which  may  be  either  diminished  by  the 
same  amount,  as  in  the  so-called  isometric,  projection,  or 
two  may  be  reduced  in  one  proportion  and  the  third  in  some 
other,  as  in  the  monodimetric  projection  ;  or,  finally,  each 
axis  may  be  reduced  in  a  different  proportion,  giving  the 
anisometric  projection.  Of  these  methods  the  first  is  not 
suited  for  representing  crystals,  as  the  distortion  of  the  faces 
is  too  great,  but  either  of  the  other  two  may  be  used,  ac- 
cording to  circumstances,  the  anisometric  projection,  as  a 
rule,  giving  the  most  natural  figures,  which,  however,  is 
accompanied  with  the  disadvantage  of  an  excessive  fore- 
shortening of  the  basal  section. 

The  following  table,  by  Weisbach,  gives  the  elements  of 
the  projections  most  commonly  used  in  this  class  of  draw- 
ing :  the  notation  refers  to  three  equal  rectangular  axes 
taken  in  Weiss's  order  ;  the  first  group  of  figures  gives  the 
proportional  lengths  assumed  for  the  axes,  the  second  the 
corresponding  reduced  lengths  compared  with  their  true 
lengths,  as  given  in  their  orthographic  projections,  and  the 
last  the  angles  between  the  axes  at  the  origin  o.  The 

O  2 


196 


Systematic  Mineralogy. 


[CHAP.  X. 


second  of  the  monodimetric  and  anisometric  series  respec- 
tively will  be  found  to  be  most  generally  suited  for  repre- 
senting crystals. 


Proportional 

Reduced  lengths  of 

Projections 

lengths  of 
axes 

axes  for  real  length 
=  1,000 

Angles  between  axes 

a      b     c 

a           be 

A  OB               AOC              HOC 

Isometric  .    . 

iii 

o'8i7    0*817    0*817 

120° 

120° 

120° 

Monodimetric 

122 

0-471    0-943    0-943 

I3i°.25; 

I3I°.25' 

97°.  n 

, 

133 

0-324    0-973    0-973 

i33°-24' 

1  33°.  24 

93°.  1  1 

r 

144 

0-246    0-985    0^985 

I34°o6' 

1  34°  06 

9i°'4/ 

Anisometric  . 

5      9    «> 

0*493    0*887    0*985 

157° 

107-49 

95°-" 

jj 

6    17    18 

o'333    0-944    0*998 

I72°.so/ 

96°-23' 

90°.  47 

, 

8    31    32 

0-250    0*968    0-999 

I74°.46' 

94-55, 

90  .20 

' 

10    49    50           0-200  :  0-980  :  i  "ooo       175.52' 

93°.  58' 

90°.  i  o' 

In  drawing  the  projection,  the  axis  c  is  made  vertical, 
and  the  other  axes  are  then  set  out  from  the  point  o,  at 
their  proper  angles,  with  a  protractor.  The  proportional 
lengths  are  then  laid  off  upon  each  side  of  the  centre,  and 
if  the  drawing  is  sufficiently  large,  these  lengths  may  be  sub- 
divided  into  equal  parts.  This  gives  the  projection,  or  '  axial 
cross,'  for  three  axes  of  the  same  length,  or  that  of  the  unit 
form  of  the  cubic  system ;  and  therefore,  by  applying  to  the 
vertical  axis  the  proper  value  of  c  as  given  in  works  on 
Descriptive  Mineralogy,  the  unit  axis  of  any  tetragonal 
species  may  be  found,  and  similarly,  by  altering  a  and  c, 
those  of  a  rhombic  species  corresponding  to  l>=  i. 

In  the  hexagonal  system  the  projections  of  the  lateral 
axes  h  and  /  are  found  from  those  of  the  cubic  system  by 
making  the  front  axis  of  the  latter  a  =  \/Y=  1732,  and 
drawing  lines  to  the  extremities  of  the  right  and  left  axis  b, 
which  in  this  case  corresponds  to  k.  This  gives  the  projec- 
tion of  a  right-angled  triangle,  or  half  the  base  of  a  rhombic 
prism,  whose  obtuse  angles  are  120°;  a  line  parallel  to  kZ, 
bisecting  a,  truncates  the  acute  angle  or  gives  a  third  side  of 
a  regular  hexagon,  and  lines  drawn  from  these  points  of  inter- 
section through  the  centre,  o,  will  be  the  projections  of  the 


CHAP.  X.]          Projection  of  Oblique  Axes.  197 

axes  required ;  the  vertical  axis  is  then  changed  to  the 
characteristic  value  of  c,  as  in  the  tetragonal  system. 

In  the  oblique  system  the  position  of  the  clinodiagonal 
axis  is  found  by  laying  off  its  projections  upon  the  rectan- 
gular unit  axes  a  and  c — that  is,  upon  the  former  a  sin.  />, 
and  upon  the  latter  c  cos.  ft — these  being  the  sides  of  a 
parallelogram  whose  diagonal  corresponds  to  the  unit-length 
of  an  axis  having  the  characteristic  inclination  required, 
the  proper  lengths  corresponding  to  the  fundamental  para- 
meters of  the  species  being  then  substituted  as  in  the 
rhombic  system. 

In  the  triclinic  system  a  similar  principle  is  followed. 
Starting  from  three  rectangular  axes  of  the  same  length,  and 
calling  the  angle  between  the  pinakoids  o  i  o  and  i  o  o, 
c,  the  diagonal  of  a  parallelogram  whose  sides  are  a  cos.  c  and 
If  sin.  c,  will  be  the  projection  of  a  horizontal  line,  b ',  whose 
length  =  i  in  the  plane  of  o  i  o.  Setting  off  upon  this,  the 
length  b'  sin.  a,  and  upon  the  vertical  c  cos.  a,1  the  sides 
of  another  parallelogram  are  obtained,  whose  diagonal  will 
be  the  projection  of  the  axis  b,  its  acute  angle  to  the  vertical 
being  to  the  right  or  left  of  the  centre  according  as  the  first 
quantity  is  taken  on  the  positive  or  negative  side  of  b',  the 
second  being  always  positive.  The  projection  of  the  front 
axis  is  found  from  the  sin.  and  cos.  of  the  angle  /3,  in  the 
manner  given  for  the  oblique  system  ;  the  third  axis,  c,  re- 
tains the  vertical  position  unchanged.  These  oblique  axes 
will  all  be  of  the  same  length,  so  that  to  obtain  those  proper 
to  the  species,  a  and  c  must  be  multiplied  by  their  character- 
istic parameters,  b  being  regarded  as  unity.  From  the  pro- 
jection of  the  axes  the  unit  form  of  a  species  is  obtained  by 
joining  their  extremities  by  straight  lines,  which,  in  the  cubic 
system,  will  represent  the  edges  of  the  octahedron  in  the 
tetragonal,  those  of  the  unit-pyramid,  and  so  on  for  the  other 
systems.  The  cube,  or,  generally,  any  diagonal  prismatic 

1  o  being  the  angle  between  the  axes  b  and  c. 


198  Systematic  Mineralogy.  [CHAP.  X. 

form,  is  found  by  ruling  through  the  extremities  of  an  axis 
lines  parallel  to  the  other  axis  in  the  same  plane,  which  meet 
in  the  section  on  the  zone  plane,  and  lines  through  the 
corners  of  this  plane  parallel  to  the  third  axis  will  be  the 
projection  of  edges  of  combination  of  faces  belonging  to 
the  zone. 

The  construction  of  forms  with  other  parameters  is 
effected  in  a  similar  manner  by  lengthening  the  unit  axes 
by  the  amounts  indicated  by  the  co-efficients  in  Weiss's 
method,  or  cutting  off  the  reciprocal  quantities,  when  working 
by  the  indices,  and  connecting  the  points  so  obtained  by 
straight  lines.  This  will  determine  a  series  of  planes  having 
the  properties  of  the  faces,  and  their  convex  portions  will 
give  the  projection  required.  This  method  of  construction 
is  shown  for  the  principal  forms  of  the  cubic  system  in 
figs.  15,  18,  25,  and  28. 

In  drawing  combinations  it  is  to  be  remembered  that  the 
octahedron,  or  unit  pyramid,  is  always  the  largest,  and  the 
cube  rectangular  prism,  or  pinakoid,  the  smallest  of  the  con- 
stituent forms.  It  is  therefore  necessary,  in  the  first  instance, 
to  determine  approximately  the  shape  of  the  latter  faces 
from  a  freehand  sketch  or  measurement  of  the  crystal  it  is 
intended  to  represent.  The  parallelepiped  formed  by  the 
pinakoids  or  analogous  faces  is  then  completed,  and  the  other 
forms  are  put  in  by  truncating  its  edges  and  angles  accord- 
ing to  their  geometrical  properties,  as  laid  down  in  works 
on  Descriptive  Crystallography.  Illustrations  of  the  methods 
to  be  followed,  with  examples,  will  be  found  in  the  larger 
text-books,  more  particularly  in  those  of  Groth  and  Dana. 
Considerable  practice,  and,  above  all,  a  knowledge  of  the  zonal 
relation  of  the  faces,  is  required,  in  order  to  obtain  the  re- 
quired results  with  the  minimum  number  of  construction  lines. 
As  a  rule  it  is  best  to  work  from  a  carefully  prepared  projection 
of  a  cube,  with  inscribed  octahedron  and  rhombic  dodeca- 
hedron, to  the  same  axes.  Upon  these  should  be  shown  the 
positions  of  the  rhombic  and  trigonal  interaxes  and  the 


CHAP.  X.]  Generalised  Projections.  199 

points  where  they  intersect  the  faces.  If  of  a  sufficient  size, 
the  derived  forms  may  be  added,  but  it  is  perhaps  better  as 
a  rule  not  to  put  more  than  two  or  three  forms  into  one 
figure,  in  order  to  prevent  confusion  by  the  multiplication  of 
lines.  A  useful  size  is  an  octahedron  of  about  six  inches 
length  of  vertical  axis.  This  should  be  drawn  upon  strong 
paper  or  cardboard,  and  used  as  a  protractor  for  projecting 
smaller  figures  by  means  of  parallel  rulers  or  set  squares 
upon  other  sheets.  In  the  other  systems  similar  projections 
for  certain  type  species  should  be  constructed.  Among  the 
most  useful  will  probably  be  found  Tinstone,  Zircon,  Ido- 
crase,  and  Apophyllite  in  the  tetragonal,  Quartz  and  Calcite 
in  the  hexagonal,  Topaz,  Barytes,  and  Sulphur  in  the  rhom- 
bic, Orthoclase  and  Augite  in  the  oblique,  and  Albite  and 
Axinite  in  the  triclinic  system.  A  hard  lead  pencil,  with  a 
fine  tapered  point,  should  be  used  in  drawing,  the  lines  being 
kept  as  fine  as  possible. 

In  all  except  the  cubic  system  the  method  of  representing 
crystals  by  projections  upon  the  plane  of  the  base,  or  that 
of  some  other  prismatic  zone,  is  attended  with  considerable 
advantage,  as  all  similar  faces  are  shown  of  similar  geome- 
trical form  and  edges  parallel  to  the  plane  of  projection  of 
their  true  lengths.  Such  projections,  however,  being  only 
horizontal  or  ground  plans,  do  not  indicate  the  shape  of  the 
prismatic  faces  of  the  zone,  which  only  appear  as  lines.  The 
best  examples  will  be  found  in  the  excellent  illustrations  to 
Brooke  and  Miller's  Mineralogy.  The  mode  of  construction 
is  that  of  ordinary  orthographic  projection. 

Generalised  representations  of  crystals.  The  representa- 
tion of  an  individual  crystal  by  either  of  the  methods  of 
projection  previously  noticed  involves  a  knowledge  of  its 
material  shape,  and  is  special  to  itself;  and  therefore,  in  those 
species  whose  series  of  forms  is  extensive,  a  great  number  of 
figures  will  be  required,  in  order  to  obtain  an  idea  of  their 
crystallographic  characters.  This  knowledge  may  be  con- 
veniently summarised  by  the  use  of  other  projections,  in 


2OO 


Systematic  Mineralogy. 


[CHAP.  X. 


which  the  faces  are  represented  by  lines  or  points  instead 
of  surfaces,  and  which  have  the  advantage  of  allowing  the 
whole  of  the  observed  faces  in  any  species  to  be  combined 
into  a  single  figure. 

In  the  first  of  these  methods,  the  linear  projection  of 
Quenstedt,  the  faces  are  represented  by  the  straight  lines, 
in  which  they  intersect  the  plane  of  projection,  supposing 
them  all  to  pass  through  a  point  in  the  normal  to  that  plane 

FIG.  316. 


at  the  distance  assumed  to  be  the  unit  axial  length.  Fig.  316 
shows  the  application  of  this  method  to  the  cubic  combina- 
tion GO  (9  GO,  O,  GO  O,  202.  The  plane  of  projection  is  the 
cube  face  (o  o  i ),  and  the  common  point  of  intersection  of  the 
faces  the  unit  length  of  the  vertical  axis,  is  represented  by 
the  central  point  c.  This  position  is  the  normal  one  for  th£ 
octahedron,  for  those  faces  of  the  rhombic  dodecahedron 
whose  indices  have  the  order  o  1 1  and  i  o  i,  and  those  of  the 
icositetrahedron  whose  order  is  112.  The  first  form,  there- 
fore, will  be  represented  by  the  projection  of  its  horizontal 
edges,  or  the  square  whose  diagonals  are  the  unit  lengths  of 


CHAP.  X.]  Linear  Projection.  2OI 

the  lateral  axes  A  and  B,  the  second  by  the  square  described 
upon  these  lengths,  and  the  third  by  the  square  whose 
diagonals  are  2  A  and  2  B.  The  remaining  faces  of  the  cube 
and  rhombic  dodecahedron,  being  parallel  to  the  vertical 
axis,  will,  when  made  to  pass  through  it,  be  represented  by 
straight  lines  intersecting  at  45°,  or  the  diagonals  of  the  pre- 
ceding squares.  The  eight  remaining  faces  of  the  icositetra- 
hedron  having  the  order  121  and  211,  intercept  the  vertical 
axis  at  2  c ;  and  therefore,  when  brought  back  to  c,  the  pro- 
jections of  the  first  four  will  be  the  four  lines  enclosing  a 
rhomb,  whose  diagonals  are  a  and  ^b,  and  those  of  the 
second  four  the  sides  of  a  similar  rhomb  having  the  dia- 
gonals ^  a  and  b.  The  zonal  relations  of  the  faces  of  the 
different  forms  entering  into  a  combination  are  indicated  in 
this  projection  by  the  crossing  points  of  the  lines,  or  the 
so-called  zonal  points.  Thus  (112)  is  the  zonal  point  of  a 
zone  containing  two  adjacent  faces  of  the  rhombic  dodeca- 
hedron, and  the  face  of  2  O  2  truncating  their  edge,  the  latter 
being  common  to  two  octants  of  the  crystal;  B  gives  the  zone 
formed  by  two  adjacent  faces  of  the  octahedron,  the  face  of 
the  rhombic  dodecahedron  truncating  their  edge  and  two 
faces  of  2  02,  and  so  on  for  many  others,  which  may  be  de- 
duced from  the  figure.  In  the  systems  of  lower  symmetry 
the  diagrams  are  less  regular,  and  apparently  less  simple,  but 
their  construction  is  easily  learnt  when  the  leading  principle 
is  mastered.  For  the  complete  description  of  these  the 
reader  is  referred  to  Quenstedt's  works  on  Crystallography 
and  Mineralogy.  An  ingenious  application  of  this  method 
in  the  construction  of  perspective  figures  of  crystals  is  given 
in  E.  S.  Dana's  'Text-book  of  Mineralogy,'  p.  427. 

Spherical  projection.  This  method,  originally  introduced 
by  Neumann,  and  brought  into  general  use  in  its  present 
form  by  Miller,  supposes  the  crystal  to  be  placed  within  a 
sphere,  both  having  a  common  centre,  and  lines  normal  to 
the  faces  to  be  drawn  through  the  centre  to  the  surface  of 
the  sphere  on  either  side.  These  lines  will  obviously  be  the 


2O2 


Systematic  Mineralogy. 


[CHAP.  X. 


diameters  of  the  sphere,  and  the  points  of  intersection  their 
poles  ;  and  as  the  angle  between  the  normals  of  two  faces 
is  equal  to  the  supplement  of  their  dihedral  angle,  the  position 
of  the  poles  will  determine  those  of  the  faces  by  a  system  of 
points  upon  the  spherical  surface,  which  give  rise  to  problems 
that  can  be  solved  by  spherical  trigonometry.  If  a  sphere 
be  supposed  to  be  projected  upon  its  equatorial  plane, 
that  being  also  parallel  to  the  terminal  pinakoid,  or  the 
equivalent  face  of  the  cube,  the  pole  of  that  face  will  be 
the  centre  of  the  circle  of  projection,  the  poles  of  faces  in 
the  zone  of  the  principal  prism  will  lie  in  the  circumference, 
their  angular  distance  being  given  directly  by  the  measured 
angles  with  the  reflecting  goniometer,  and  those  of  the 


FIG.  318. 


pyramid,  or  other  inclined  forms,  will  be  at  various  intermediate 
points,  all  being  so  related  that  the  poles  of  all  the  faces  in 
a  zone  will  lie  upon  the  same  great  circle,  which,  from  the 
property  of  the  projection,  will  either  be  a  straight  line  when  it 
passes  through  the  pole  of  the  circle  of  projection,  or  an  arc  of 
a  circle  when  in  any  other  position.  Fig.  317  is  an  example  of 
the  spherical  projection  of  a  crystal  of  Topaz,  containing  the 


CHAP.  XT.]  SpJierical  Projection.  203 


forms  oo  P,  00/2,  P,  ±  P,  ip,  f  As,  /oo,  ocAo,  o  P, 
as  shown  in  horizontal  projection  in  fig.  318.  In  it  the 
principal  zones,  or  those  that  include  the  terminal  pinakoid, 
are  represented  by  diameters,  which  are  projections  of 
meridians  of  the  sphere,  whose  direction  is  found  by  laying 
off  at  the  centre  the  supplements  to  the  interfacial  angles 
of  the  prism  from  the  pole  of  o  i  o  ;  the  position  of  the 
pole  of  any  intermediate  face  in  these  zone  circles  being 
found  by  taking  in  them  a  distance  equal  to  the  tangent  of 
half  the  supplement  ot  the  inclination  of  the  face  upon 
o  P,  the  radius  being  considered  as  unity.  Zones  that  are 
not  perpendicular  to  the  plane  of  projection,  or  do  not 
include  o  P,  are  represented  by  arcs  of  circles,  these  being 
the  stereographic  projections  of  great  circles  oblique  to 
the  equatorial  plane.  This  system  has  the  advantage  of 
representing  the  faces  of  crystals  in  the  most  general  manner 
—  namely,  by  points  —and  therefore  there  is  no  limit  to  the 
number  of  them  that  can  be  included  in  a  single  figure. 
Some  very  remarkable  examples  will  be  found  in  Descloi- 
zeaux's  '  Text-book  of  Mineralogy.' 


CHAPTER   XL 

PHYSICAL   PROPERTIES   OF   MINERALS  : — CLEAVAGE, 
HARDNESS,  SPECIFIC   GRAVITY,  ETC. 

THE  methods  employed  in  the  second  great  division  of 
mineralogical  research,  or  the  investigation  of  the  struc- 
tural peculiarities  of  minerals,  are  essentially  those  of  experi- 
mental mechanics  and  physics.  All  such  investigations  are 
based  upon  the  assumed  existence  of  physical  molecules  or 
indivisible  particles  of  matter,  all  of  the  same  kind  and 
similarly  arranged  in  the  same  substance.  Without  entering 
into  the  question  of  the  actual  nature  of  such  molecules,  or 


204  Systematic  Mineralogy.  [CHAP.  XI. 

atoms,  it  appears  to  be  certain  that  a  great  part  of  the 
results  of  observations  may  be  explained  by  assuming  that 
the  molecular  centres  occupy  the  points  of  a  reticular 
system,1  and  that  the  molecules  may  be  considered  as  con- 
centrated upon  such  points  as  attractive  or  repulsive  centres 
of  force.  As  a  working  hypothesis  this  involves  the  further 
supposition  that  if  they  are  to  be  regarded  as  having  definite 
size  or  dimensions  in  space,  these  dimensions  must  be  very 
small  as  compared  with  the  linear  distance  between  any 
two.  By  the  substitution  of  such  centres  of  force  for  geo- 
metrical points  in  the  reticular  systems  of  the  crystallo- 
grapher,  we  arrive  at  the  idea  of  the  physical  as  distinguished 
from  the  morphological  crystal,  in  which  the  consideration 
of  molecular  arrangement  precedes  that  of  form,  the  latter 
being  a  consequence  of  such  arrangement.  These  investi- 
gations are  especially  applicable  to  those  minerals  whose 
crystals  appear  in  forms  common  to  two  or  more  systems, 
such  as  the  combination  of  three  pairs  of  rectangular 
planes,  or  the  regular  six-sided  prism,  the  true  character 
of  which  can  only  be  determined  by  a  knowledge  of  their 
physical  characteristics. 

Without  anticipating  what  will  subsequently  be  treated 
in  detail,  it  may  be  generally  stated  that  the  observations 
upon  which  conclusions  as  to  molecular  structure  are  founded 
are  essentially  those  of  elastic  resistance  to  forces  tending  to 

1  There  are  two  conjugate  forms  of  the  relation  between  the 
molecular  centres  and  the  points  of  a  reticulation.  They  may  coincide, 
or  the  molecular  centres  may  occupy  the  centres  of  gravity  of  the 
(tridimensional)  mesh.  In  a  strictly  cubical  system  this  merely  shifts 
the  position  in  space  of  the  whole  reticulation.  But  it  is  quite  different 
in  the  well-known  case  of  round  shot  in  pile.  The  centres  of  these  lie 
on  the  knots  of  a  fourfold  reticulation,  dividing  space  into  octahedra 
and  tetrahedra  in  the  numerical  ratio,  I  :  2,  parallel  to  the  faces  of  a 
regular  tetrahedron  ;  but  their  bounding  planes  form  a  network,  each 
mesh  of  which  is  a  right  rhombic  dodecahedron.  Some  writers  use 
one,  and  some  the  other,  hypothesis. 


CHAP.  XL]  Cleavage.  2O$ 

alter  the  equilibrium  of  the  mass,  including  under  this  head 
the  coarser  mechanical  agency  necessary  to  produce  fracture, 
as  well  as  the  more  subtle  evidence  derived  from  the  trans- 
mission of  vibratory  movements,  as  manifested  in  the  action 
of  light,  heat,  electricity,  and  magnetism.  In  some  cases 
we  find  that  the  effects  of  these  agents  are  similar  in  any 
direction,  while  in  others  there  is  decided  dissimilarity  in 
different  directions,  and  these  differences  are  intimately  con- 
nected with  crystalline  symmetry.  In  addition  to  these 
there  are  two  properties  common  to  all  solids  alike,  and 
therefore  independent  of  structure — namely,  density  and 
hardness,  or  specific  cohesive  power — which  are  of  value  in 
Determinative  Mineralogy,  but  it  will  be  more  convenient  to 
consider  first  the  structural  property  where  relation  to  crys- 
talline form  is  most  apparent. 

Cleavage.  The  resistance  opposed  by  cohesion  in  homo- 
geneous amorphous  substances  to  forces  tending  to  displace 
or  separate  their  particles  is  similar  in  any  direction,  so  that 
there  is  no  reason  why  they  should  yield  more  readily  in 
one  way  than  another  when  strained  beyond  their  elastic 
limits  by  forces  of  any  kind,  and  therefore,  when  so  treated 
they  will  break  up  into  fragments  of  essentially  irregular 
form.  It  is,  however,  different  in  the  case  of  crystals,  which, 
in  a  large  number  of  instances,  when  similarly  treated,  show 
very  decided  tendencies  to  separate  into  fragments  bounded 
by  planes  which  are  for  the  most  part  related  in  some  simple 
manner  to  the  unit  form  of  the  series  proper  to  the  substance ; 
and  when  the  direction  of  such  surfaces  is  known,  a  com- 
paratively slight  cutting  or  wedging  strain  will  be  sufficient 
to  produce  a  separation,  while  the  resistance  in  other  direc- 
tions may  be  considerably  greater.  This  property  is  known 
as  crystalline  cleavage,  and  the  surfaces  of  separation  are 
called  cleavage  planes.  It  is  directly  related  to  crystalline 
structure,  but  has  no  relation  to  specific  cohesive  power  or 
hardness  as  measured  by  the  resistance  to  abrasion,  the 
hardest  known  mineral,  Diamond,  being  one  of  the  most 


2o6  Systematic  Mineralogy.  [CHAP  XL 

commonly  cleavable,  as  is  also  Gypsum,  one  of  the  softest ; 
while  among  those  of  intermediate  hardness,  Quartz,  Garnet, 
and  Pyrites,  are  scarcely  cleavable,  but  break  like  masses  of 
glass  or  other  amorphous  bodies. 

The  readiest  way  of  determining  the  cleavage  of  a  crystal 
is  to  place  the  edge  of  a  knife  or  small  chisel  upon  a  face 
parallel  to  that  of  some  principal  form,  and  strike  a  light 
blow  with  a  hammer,  when,  if  the  direction  is  near  that  of  a 
principal  cleavage,  a  more  or  less  flat-faced  fragment  will  be 
removed.  If,  on  the  other  hand,  no  cleavage  is  obtainable 
in  the  direction  of  the  blow,  the  fractured  surface  will  be 
uneven  and  irregular,  or  will  show  traces  of  step-shaped 
structure  in  the  direction  of  the  true  cleavage  plane.  Thus 
Fluorspar,  whose  crystals  are  chiefly  cubes,  cannot  be  cleaved 
parallel  to  that  form,  but  yields  with  the  greatest  ease  in  the 
direction  of  an  octahedral  face.  Galena  has  an  extremely 
perfect  cleavage  parallel  to  the  faces  of  the  cube ;  and  Zinc- 
blende  to  those  of  the  rhombic  dodecahedron.  Some 
minerals,  such  as  Mica  and  Gypsum,  are  very  easily  cleavable, 
and  may  with  slight  effort  be  divided  by  the  finger-nail,  or 
the  point  of  a  knife,  or  needle,  into  lamina?  of  extreme  thin- 
ness. In  the  case  of  Mica  there  seems  to  be  no  limit  to  the 
capacity  for  cleavage,  as  laminae  may  be  obtained  thinner 
than  the  edge  of  any  cutting  tool  that  can  be  brought  to 
bear  upon  them. 

In  some  instances  cleavages  may  be  developed  in  im- 
perfectly cleavable  crystals  by  strongly  heating  and  sud- 
denly cooling  them  in  water.  Quartz  crystals,  when  so 
treated,  occasionally  develop  faces  parallel  to  those  of  the 
unit  rhombohedron  ;  and  under  ordinary  circumstances  they 
break  with  a  fracture  like  that  of  glass.  Easily  cleavable 
minerals,  such  as  Salt,  Galena,  Fluorspar,  and  Calcite,  usually 
decrepitate,  or  fly  to  pieces,  when  suddenly  heated,  the  frag- 
ments obtained  being  regular  cleavage  forms. 

In  the  cubic  system,  where  all  three  axes  are  physically 


CHAP.  XL]  Cleavage.  207 

equivalent,  a  cleavage  parallel  to  any  face  of  a  form  requires 
the  existence  of  a  similarly  perfect  one  parallel  to  the  remain- 
ing faces,  and  therefore  the  cleavage  form  will  be  a  closed 
one,  as  are  also  the  forms  produced  by  pyramidal  and  rhom- 
bohedral  cleavage  in  the  tetragonal  and  hexagonal  systems  ; 
but  the  prismatic  and  basal  cleavage  in  these  systems,  as  well 
as  all  those  in  the  remaining  systems,  can  only  give  open 
forms,  and  therefore,  to  obtain  regular  fragments,  a  cleavage 
in  two  or  more  directions  is  required  in  the  systems  of 
lower  symmetry.  As,  however,  these  are  not  always  found, 
or,  when  present,  are  of  very  unequal  value,  the  general 
cleavage  tendency  of  crystals  belonging  to  these  systems  is 
to  produce  parallel  plates  from  the  extreme  development 
of  a  principal  cleavage  as  compared  with  others.  This  is 
very  strongly  marked  in  the  Mica  group,  whose  crystals  are 
cleavable  without  limit  parallel  to  the  base,  but  are  exceed- 
ingly tough  and  strong  in  other  directions.  In  describing 
minerals  with  two  or  more  cleavages,  it  is  necessary,  there- 
fore, to  indicate  their  quality  as  well  as  their  direction.  The 
terms  used  for  this  purpose  are  highly  perfect \  as  in  Mica;  very 
perfect,  as  in  Fluorspar,  Barytes,  and  Hornblende  ;  perfect, 
as  in  Augite  and  Chrysolite  ;  imperfect,  as  in  Garnet  and 
Quartz  ;  and  very  imperfect,  when  only  traces  of  cleavage 
can  be  obtained. 

The  following  are  the  principal  directions '  of  cleavage 
observed  in  the  different  systems,  with  a  few  examples  of 
each,  from  which  it  will  be  seen  that  only  the  simpler 
forms,  or  those  with  low  indices,  are  possible  as  cleavage 
forms  : — 

i.   Cubic  system. 

Octahedral  :  Fluorspar,  Sal-ammoniac,  Diamond. 
Cubical :   Common  Salt,  Galena. 
Rhombic  dodecahedral  :  Zincblende. 


2o8  Systematic  Mineralogy.  [CHAP.  XI. 


2.  Hexagonal  system. 

Pyramidal  of  either 

order          .         .         P,  P2          Pyromorphite. 
Prismatic  of  either)    00oo0^>2   j  Apatite,  Red-zinc  Ore, 

order          .         .  [  (     Cinnabar. 

Basal     .         .        .  o  P  Beryl,  Red-zinc  Ore. 

Rhombohedral       .  Jt  The    Carbonates    of 

the  Calcite  group, 
Quartz. 

3.  Tetragonal  system. 

Pyramidal  of  either)        „     p         (Scheelite,  Copper  Py- 

order         .         .)  {     rites. 

Prismatic  of  either 

order         .         .  GO  P,  oo  Pec     Rutile,  Tinstone. 
Basal    ...  oP  Apophyllite. 

4.  Rhombic  system. 

Prismatic  .  .  ecP  White-lead  Ore,  Na- 

trolite. 

Macrodomatic       .  Pec  Barytes. 

Brachydomatic      .  Pec  Barytes. 

Basal    ...  oP  Topaz,  Prehnite. 

Macropinakoidal  .         ccPeo          Anhydrite. 

Brachypinakoidal .  oo  Pco  Barytes,  Antimony 

Glance. 

5.  Oblique  system. 

Hemipyramidal     .  ±  P           Gypsum. 

Prismatic       .         .  oo  P          Hornblende,  Angite. 

Clinodomatic         .  -Pec           Azurite. 

Hemiorthodomatic  +Pec         Euclase. 

Basal     ...  oP           Orthoclase,     Epidote, 

Mica. 

Orthopinakoidal    .  oo  ^oo          Epidote. 

Clinopinakoidal     .  oo  iPoo         Orthoclase,  Gypsum. 


CHAP.  XL]  Fracture.  209 

6.   Tridinic  system. 

Hemiprismatic       .  oo  F  oo  'P  Labradorite. 

Hemidomatic  'P  GO  Cryolite. 

Basal    .         .         .  oP  Triclinic  felspar  group. 

Macropinakoidal  .  oo  /'oo  Cyanite, 

Brachypinakoidal .  oo  P<x>  Axinite. 

Besides  the  cleavage  surfaces,  other  divisional  planes 
indicating  structure  may  be  obtained  in  some  minerals  by 
special  treatment.  Thus,  by  filing  away  two  opposite  edges 
of  a  cube  of  Salt  parallel  to  the  face  of  a  rhombic  dodeca- 
hedron, and  screwing  it  up  carefully  in  a  vice,  an  internal 
fracture  parallel  to  a  face  of  the  latter  form  will  be  produced. 
A  similar  face  parallel  to  —  i  R  may  also  be  readily  pro- 
duced in  Calcite.  Another  method  consists  in  the  use  of 
a  conical  steel  point  such  as  a  centre  punch,  which,  when 
placed  on  the  face  of  a  crystal  and  struck,  often  develops  a 
system  of  cracks  parallel  to  some  principal  crystallographic 
directions. 

Fracture.  The  characteristic  appearances  of  the  sur- 
faces of  minerals,  broken  in  directions  that  are  not  cleavage 
planes,  are  described  by  the  following  terms  : — 

i.  Conchoidal.  This  is  the  characteristic  fracture  of  homo- 
geneous amorphous  substances,  the  surfaces  presenting  an 
alternation  of  rounded  ridges  and  hollows  ;  it  is  best  seen  in 
glass  and  imperfectly  cleavable  minerals,  such  as  Quartz, 
Garnet,  and  Ice.  According  to  the  nature  of  the  undula- 
tions of  the  surface,  it  is  further  characterised  as  flat-  or  deep-, 
coarsely  or  finely,  conchoidal. 

2.  Smooth,  when  the  surface,  without  being  absolutely 
plane,  presents  no  marked  irregularities. 

3.  Splintery,  when  the  surface  is  covered  with  partially 
separated  splinters  in  irregular  fibres,  as  in  fibrous  Hematite. 

4.  Hackly,    when   the   surface  is  covered  with  ragged 


2io  Systematic  Mineralogy.  [CHAP.  XI. 

points  and  depressions.  This  is  especially  characteristic  of 
native  metals. 

In  easily  cleavable  minerals  it  is,  as  a  rule,  difficult  to 
develop  any  special  fracture,  but  it  may  sometimes  be  done 
by  striking  a  fragment  a  sharp  blow  with  a  blunt  point,  as 
that  of  a  rounded  hammer  or  pestle,  when  traces  of  charac- 
teristic fractures  may  occasionally  be  obtained,  springing 
across  from  one  cleavage  surface  to  another. 

Hardness.  By  this  term  is  meant  the  resistance  of  the 
surface  of  a  mineral  to  abrasion  when  a  pointed  fragment  of 
another  substance  is  drawn  rapidly  across  it  without  sufficient 
pressure  to  develop  cleavage  separation.  When  the  latter 
is  the  harder  substance  it  will  scratch  the  former,  but  when 
it  is  the  softer  the  point  will  be  blunted  without  the  surface 
passed  over  being  affected.  As  there  are  very  considerable 
differences  between  minerals  in  regard  to  this  test,  it  is  one 
of  the  most  important  of  their  physical  constants;  but  as  there 
is  no  means  of  expressing  the  results  in  absolute  measure, 
recourse  must  be  had  to  an  indirect  method,  in  which  com- 
parative hardness  is  measured  by  a  scale  of  typical  minerals. 
This,  known  as  Moh's  scale  of  hardness,  is  as  follows  : — 

1.  Talc  6.  Orthoclase. 

2.  Gypsum  or  Rock  Salt  7.  Quartz. 

3.  Calcite.  8.  Topaz. 

4.  Fluorspar.  9.  Corundum. 

5.  Apatite.  10.  Diamond. 

Breithaupt.  while  preserving  these  numbers,  proposed  to 
interpolate  Mica  as  2.5  and  Scapolite  as  5.5,  but  they  have 
not  been  generally  adopted. 

The  softer  numbers  of  the  scale,  Talc  and  Gypsum,  may 
be  scratched  by  the  finger  nail,  and  those  up  to  6  by  a  file 
or  the  point  of  a  knife,  while  Quartz  and  the  higher  numbers 
are  all  harder  than  steel. 

In  testing  a  mineral  for  hardness,  it  is  applied  successively 
to  different  numbers  of  the  scale  until  one  is  found  that  can 


CHAP.  XI.]  Hardness.  211 

be  scratched  by  it.  The  positions  are  then  reversed,  and  if 
it  is  found  that  the  scale  mineral  will  scratch  that  under  ex- 
amination, both  are  said  to  be  of  the  same  hardness ;  but  if 
it  is  not  scratched,  its  hardness  is  said  to  be  intermediate 
between  that  of  the  particular  number  and  the  next  harder 
one.  Thus,  the  hardness  of  Barytes,  which  scratches  but  is 
not  scratched  by  Calcite,  and  is  easily  scratched  by  Fluor- 
spar, is  said  to  be  3.5,  or  between  3  and  4.  Small  differences 
in  the  hardness  of  substances  may  also  be  appreciated  by 
drawing  fragments  of  about  the  same  size  over  a  flat  file, 
when  the  harder  substance  will  give  a  sharper  sound  than 
the  softer  one  ;  and  if  they  are  both  of  lower  hardness  than 
quartz,  the  softer  one  will  leave  the  largest  amount  of  powder 
or  streak  on  the  file. 

The  hardness  of  crystals  that  are  easily  cleavable  often 
shows  very  decided  differences  in  different  directions.  This 
subject  has  been  investigated  by  Exner,  who  found  that  the 
minimum  load  upon  a  steel  point  necessary  to  produce 
abrasion  when  moving  over  a  cube  of  rock  salt  parallel  to 
the  diagonal  of  the  face  was  one-third  greater  than  that 
required  when  the  line  of  motion  was  parallel  to  an  edge. 
Similarly  the  face  of  a  rhombohedron  of  Calcite  may  be 
more  easily  scratched  parallel  to  the  principal  cleavage  than 
in  any  other  direction.  It  is  better,  therefore,  in  making  up 
a  scale  of  hardness,  to  use  irregularly  broken  fragments, 
when  they  can  be  obtained  of  sufficient  purity,  rather  than 
perfect  crystals,  but  when  the  latter  are  used  the  trial 
should  be  repeated  with  different  faces. 

For  cabinet  or  indoor  use  a  complete  scale  made  up  of 
a  few  pieces  of  each  number,  with  the  possible  exception  of 
the  diamond,  should  be  arranged  in  a  divided  box,  together 
with  a  flat  file  not  too  coarsely  cut ;  but  for  travelling  use 
the  numbers  3  to  8  will  be  found  sufficient  in  most  cases. 
The  nail-trimming  blade  of  a  penknife  is  also  extremely 
useful  in  trying  the  hardness  of  minerals. 

In  testing  cut  gems,  care  is  required,  especially  with  those 


212  Systematic  Mineralogy.  [CHAP.  XL 

of  doubtful  authenticity,  not  to  disfigure  them,  and  it  is 
therefore  best  in  the  first  instance  to  apply  the  test  of  the  file 
cautiously  to  the  border  of  the  specimen,  where  a  scratch 
may  be  hidden  by  the  setting  when  the  stone  is  mounted. 

Tenacity.  This  term  is  rather  loosely  applied  in  mine- 
ralogy to  the  behaviour  of  minerals  when  subjected  to  the 
action  of  cutting  or  pulverising  tools,  the  different  degrees 
being  distinguished  as  follows  : — 

1.  Brittle:    The   substance   breaks   up   into  fragments 
without  extending.     By  far  the  larger  number  of  minerals 
are  of  this  kind  and  no  special  examples  need  be  given. 

2.  Ductile  :  The  substance  can  be  cut  with  a  knife,  but 
crushes  to  powder  under  a  hammer.     The  property  is  seen 
in  Copper  Glance  and  Copper  Pyrites,  and  in  a  higher  de- 
gree in  Silver  Glance   and  Chloride  of  Silver;   the  latter 
cuts  into  shavings  like  horn,  and  can  scarcely  be  powdered. 

3.  Malleable  :   The  substance  can  be  cut  into  shavings 
and  beaten  out  under  the  hammer.     The  soft  native  metals, 
Copper,  Silver,  and  Gold,  are  examples. 

4.  Flexible:     The   substance  when   divided  into   thin 
plates  can  be  bent,  and  remain  so  without  breaking,  as 
Talc. 

5.  Elastic:    A  thin  plate  of  the  substance  when  bent 
springs  back  to  its  original  form  when  the  strain  is  removed. 
This  happens  with  Mica  plates  that  are  not  too  thin.     An- 
other remarkable  example  is  the  elastic  Bitumen,  or  Elaterite 
of  Derbyshire. 

Although  not  a  common  property,  a  few  instances  of 
extreme  toughness  are  known  in  minerals,  such  as  certain 
varieties  of  Serpentine,  the  massive  Felspars,  Jade,  and 
Saussurite.  These  are,  as  a  rule,  uncrystalline  substances, 
and  their  toughness  is  in  no  relation  to  their  hardness. 
Malleable  native  Copper,  especially  when  intimately  mixed 
with  siliceous  vein  stuff  and  some  varieties  of  Hematite  and 
Iron  Pyrites,  has  the  same  property  in  a  high  degree. 
Probably  the  uncrystalline  diamond  or  carbonado  may  be 
considered  as  combining  toughness  with  extreme  hardness. 


CHAP.  XL]  Density.  213 

Density  and  Specific  Gravity.  The  density  of  a  substance 
is  the  mass  of  its  unit  volume  expressed  in  units  of  weight. 
Specific  gravity  is  the  ratio  of  the  density  of  a  substance  to 
that  of  another  assumed  as  a  standard,  or  of  the  weights  of 
equal  volumes  of  both.  Water  at  the  ordinary  temperature 
of  air,  or  at  that  of  its  maximum  density,  is  usually  adopted 
as  the  standard  substance.  In  the  metrical  system  the  same 
number  expresses  the  weight  of  a  cubic  centimetre  in 
grammes,  or  both  density  and  specific  gravity. 

The  term  '  density '  is  generally  used  by  French  minera- 
logists, and '  specific  gravity '  by  those  of  England  and  Ger- 
many, in  describing  minerals.  Rammelsberg,  however,  em- 
ploys the  equivalent  expression  '  volume-weight.' 

Specific  gravity  is  one  of  the  most  important  factors  in 
determinative  mineralogy,  as  it  is  found  to  be  constant  or 
variable  within  small  limits  in  different  varieties  of  the  same 
species,  while  the  differences  between  different  species  is 
often  very  considerable,  the  observed  range  being  from 
0.75 — 0.90  in  some  liquid  hydrocarbons,  to  21  or  22  in 
the  metals  of  the  platinum  group.  It  will,  however,  be 
seen  in  examining  a  classified  list  of  the  specific  gravities  of 
minerals,  such  as  those  published  by  Websky  and  the 
Bureau  de  Longitude,1  that  species  of  analogous  composition 
and  constitution  are  generally  near  together,  and  that  upon 
this  characteristic  a  certain  rough  grouping  is  possible,  as  in 
the  following  examples  : — 

o'5 — 1.5.  Most  fossil  resins,  Petroleum  and  Bitumen, 
i.o — 2.0.  Coal  Lignite  and  carbonaceous  minerals  gene- 
rally, many  hydrated  Alkaline  Sulphates  and 
Borates,  Nitre. 

2.0 — 2.5.  Sulphur,  Graphite,  Nitrate  of  Sodium,  Salt,  Gyp- 
sum, and  most  Zeolites. 
2.5 — 2.75.  Quartz,  the  Felspars,  Talc,  Serpentine,  Calcite 

Emerald. 

1  These,  with  other  admirable  tables  of  the  physical  constants  of 
minerals,  should  be  in  the  possession  of  every  student.  They  are 
contained  in  the  Anmtaire  for  1876  and  later  years. 


214  Systematic  Mineralogy.  [CHAP.  XI. 

2.8 — 3.0.  Aragonite,    Magnesite,    Dolomite,    Tremolite, 

Wollastonite,  Mica. 

3.0 — 3.5.  Apatite,  Fluorspar,  Epidote,  the  Pyroxene  and 

Hornblende   groups,  Tourmaline,    Olivine, 

Axinite,  Diamond,  Topaz,  the  Sulphides  of 

Arsenic. 

3.5 — 4.0.  Spinel,  Corundum,  Siderite,  Limonite,  Strontia- 

nite,  Celestine. 
4.0 — 4.5.  Rutile,  Zircon,  Barytes,  Witherite,  Zincblende, 

Copper  Pyrites,  Magnetic  Pyrites. 

4.5 — 5.5.  Hematite,  Magnetite,  Iron  Pyrites,  the  Ruby 
Silver  Ores,  and  many  compound  sulphides 
not  containing  Lead. 

5.6 — 6.6.  Arsenic,  Arsenical  Pyrites,  Oxides  of  Manga- 
nese, and  many  compound  sulphides  contain- 
ing Lead  and  Silver. 

6.7 — 7.9.  Antimony,   Sulphide  and  Carbonate  of  Lead, 
Sulphide  of  Silver,  Tinstone,  and  Pitchblende. 
8.0 — 9.0.  Tellurides  of  Gold  and  Silver,  Sulphide  of  Mer- 
cury, and  Copper. 

9.0 — i  i.o.  Bismuth,  Silver,  and  Palladium. 
12.0 — 15.0.  Mercury,  and  Amalgams. 
15.0 — 20.0.  Gold,  Platinum,  and  several  of  its  associated 

metals. 
Above  20.  The  native  alloys  of  Osmium  and  Iridum. 

Metallic  Iridum,  when  perfectly  purified  from  the  metals 
with  which  it  is  usually  associated,  is  the  densest  substance 
known,  having  a  specific  gravity  of  22.4. 

The  determination  of  specific  gravity  is  in  principle 
very  simple,  the  substance  being  first  weighed  in  air,  and 
then  in  water,  the  difference  between  the  two  weights  gives 
the  weight  of  an  equivalent  volume  of  water,  and  the  quo- 
tient of  the  original  weight  by  the  difference  will  be  the 
specific  gravity.  An  exact  determination  is,  however,  a 
matter  of  considerable  nicety,  and  involves  the  use  of  deli- 
cate balances,  such  as  are  only  found  in  laboratories.  The 


CHAP.  XL]  Specific  Gravity.  215 

general  details  of  manipulation  will  be  found  in  the  larger 
treatises  on  practical  chemistry.  When  the  substance  con- 
tains cavities,  it  is  necessary  to  powder  it  before  taking  the 
specific  gravity,  and  in  such  cases  the  determination  is  made 
by  estimating  the  amount  of  water  displaced  by  a  known 
weight  of  the  powder  from  a  bottle  that  has  been  exactly 
filled  with  water  and  weighed  at  a  standard  temperature. 

As  most  minerals  enclose  more  or  less  of  hollow  or  air- 
filled  spaces,  it  is  not  remarkable  that  they  should,  as  a  rule, 
be  denser  when  powdered  than  in  the  solid  state.  In 
irregular  aggregates  the  difference  is  often  very  considerable. 
Thus,  the  spongy  substances  known  as  float-quartz  and 
pummice  in  masses  are  apparently  lighter  than  water  from 
the  large  amount  of  air  enclosed,  although  when  pulverised 
their  true  density  is  found  to  be  between  two  and  three 
times  as  much. 

In  the  determination  of  the  specific  gravity  of  mineral 
masses  of  large  size  and  known  weight,  the  method  of 
gauging  the  volume  of  water  displaced  may  be  conveniently 
used.  One  of  the  best  is  that  given  by  Mohr,  which  is  sus- 
ceptible of  considerable  accuracy.  The  gauging  vessel  is  a 
glass  cylinder,  which  is  filled  with  water  to  a  standard  point 
formed  by  a  needle  projecting  from  a  slip  of  wood  across  the 
top,  the  exact  level  being  attained  when  the  front  of  the 
needle  and  its  reflected  image  in  the  water  coincide.  The 
weighed  substance  is  then  carefully  lowered  into  the  cistern, 
when  it  displaces  its  own  volume  of  water,  with  a  correspond- 
ing rise  of  the  surface  level.  The  amount  of  displacement 
is  measured  by  drawing  the  water  into  a  graduated  tube  or 
burette  until  the  original  level  is  restored.  A  convenient 
size  of  graduated  tube  is  the  ordinary  alkalimeter  used  in 
volumetric  analysis  containing  1,000  grains,  and  divided  into 
5 -grain  spaces,  or  an  equivalent  one  with  metrical  divisions. 
The  level  of  the  water  may  be  adjusted  with  great  nicety  by 
a  simple  valve  formed  of  a  piece  of  glass  rod  inserted  in  the 
indiarubber  delivery  tube,  the  aperture  of  which  can  be 


216  Systematic  Mineralogy.  [CHAP.  XI. 

varied  by  slight  pressure  of  the  finger  upon  the  tube.  This 
method,  which  has  the  advantage  of  not  requiring  a  correc- 
tion for  temperature,  is  well  adapted  for  taking  the  specific 
gravities  of  Coal.  Limestone,  and  similar  substances  ranging 
from  2  to  3,  which  can  be  used  in  fragments  of  about  half  a 
pound  weight. 

Jolly's  spring  balance,  another  co  /ance  for  the  ap- 
proximate determination  of  specific  gravity,  is  recommended 
as  being  expeditious  in  use,  fairly  accurate,  and  dispensing 
entirely  with  the  determination  of  absolute  weight.  It  con- 
sists essentially  of  a  pair  of  scale  pans  suspended  one  above 
the  other  ;  the  upper  one  is  attached  to  the  end  of  a  coiled 
steel  spring,  and  the  lower  one  is  immersed  in  a  cistern  of 
water  standing  on  a  bracket,  whose  position  can  be  adjusted 
by  a  sliding  movement  worked  by  a  rack  and  pinion.  The 
face  of  the  upright  bar  to  which  the  arm  carrying  the  spring 
is  attached  has  a  silvered  mirror  with  a  scale  of  equal  parts 
engraved  upon  it  fixed  to  it  in  front,  and  the  upper  scale 
pan  carries  a  pointer.  When  in  use  the  level  of  the  water 
vessel  is  so  adjusted  that  the  lower  scale  pan  may  be  freely 
immersed  when  a  reading  is  taken  by  bringing  the  pointer 
into  coincidence  with  its  reflected  image  in  the  glass  scale. 
The  mineral  is  then  placed  in  the  upper  scale,  whereby  the 
spring  is  distended  to  a  certain  point,  which  is  determined 
by  a  second  reading,  which  when  deducted  from  the  first 
measures  the  weight  in  air.  By  removing  it  to  the  immersed 
pan,  the  strain  is  diminished,  and  the  pointer  consequently 
rises  to  an  amount  determined  by  a  third  reading,  which 
when  subtracted  from  the  second  measures  the  loss  of 
weight  in  water.  It  is  of  course  essential  to  accuracy  that 
the  spring  should  deflect  equally  for  equal  weights  applied, 
or  it  must  not  be  strained  to  anywhere  near  its  elastic  limits, 
while  at  the  same  time  it  must  be  sufficiently  free  to  work  to 
move  through  a  measurable  distance  by  a  moderate  change 
of  weight.  The  instrument,  which  is  very  highly  spoken  of 
by  Von  Kobell,  is  made  at  Munich  at  a  cost  of  24^. 


CHAP.  XL]  Sonstedts  Method.  217 

The  density  of  large  masses  of  an  approximately  regular 
figure  may  be  roughly  determined  by  weighing  them  and 
calculating  their  cubic  volume  from  their  measured  dimen- 
sions. The  specific  gravity  is  found  by  dividing  the  weight 
by  the  contents  in  cubic  feet  multiplied  by  62.4  Ibs.,  or  the 
weight  of  a  cubic  foot  of  water.  The  reverse  operation  of 
calculating  the  weight  of  a  measured  mass  from  the  known 
specific  gravities  of  its  components  is  often  useful  on  the 
large  scale  in  estimating  the  probable  yield  of  mineral  de- 
posits, and  the  student  may  therefore  be  recommended  to 
become  familiar  with  it  by  practice. 

Specific  gravity  is  in  another  class  of  approximate 
methods  determined  by  immersion  in  a  fluid  of  known 
density,  when  the  substance,  if  heavier,  will  sink,  but  if 
lighter,  it  will  float.  This  is  specially  useful  in  the  discrimina- 
tion of  cut  stones  ;  the  fluid  used  for  this  purpose,  called 
after  the  inventor,  Sonstedt's  solution,  is  made  by  saturating 
a  solution  of  iodide  of  potassium  in  water  with  iodide  of 
mercury,  which  gives  a  liquid  having  a  maximum  density  of 
2.77,  and  which,  when  diluted  with  water,  mixes  without 
sensible  change  of  volume,  and  therefore  the  specific  gravity 
is  proportional  to  the  amount  of  the  two  fluids  in  the  mix- 
ture. This  is  also  useful  as  a  means  of  separating  mixed 
minerals  for  analysis,  when  they  are  so  intimately  associated 
as  to  be  incapable  of  separation  by  hand ;  as,  for  instance, 
the  Felspars  and  lighter  silicates  in  a  rock  may  be  roughly 
divided  from  the  denser  ferrous  and  magnesian  ones  by  a 
solution  of  a  specific  gravity  of  about  2.75,  when  the  first 
mineral  will  just  float,  while  the  latter  will  sink  readily.  The 
chief  drawback  to  the  use  of  this  substance  is  in  its  ex- 
tremely poisonous  character,  and  it  can  therefore  be  scarcely 
recommended  except  for  laboratory  use. 

An  extension  of  the  same  principle  adapted  for  the 
separation  of  the  denser  constituents  of  mineral  sands  has 
been  proposed  by  Breon ,  who  uses  molten  chloride  of  zinc 
or  lead,  or  mixtures  of  both.  These  give  fluids  whose 


2i8  Systematic  Mineralogy.  [CHAP.  XII. 

density  ranges  from  3.0  to  5.0.  As  these  salts  can  be 
melted  in  glass  tubes,  and  are  alike  soluble  in  water,  the 
separated  substances  can  be  obtained  in  the  pure  state  with- 
out much  trouble.  They  have  been  applied  in  the  separa- 
tion of  Tinstone,  Rutile,  Magnetite,  and  other  heavy  minerals 
from  Quartz,  and  silicates  in  the  microscopic  investigation 
of  sands. 

The  same  method  is  applied  on  the  large  scale  in  the 
separation  of  Gold  from  Galena  and  Iron  Pyrites  by  a  fluid 
of  intermediate  density,  namely  Mercury,  in  the  Hungarian 
Gold-mill,  although  in  this  case  the  result  is  not  quite  so 
simple,  the  metal  being  to  some  extent,  at  any  rate,  dissolved 
in  the  separating  fluid. 


CHAPTER  XII. 
OPTICAL  PROPERTIES  OF  MINERALS. 

UNDER  this  head  are  included  the  general  and  more  apparent 
properties  of  colour,  lustre,  and  translucency  or  opacity, 
which  are  common  to  all  minerals  alike,  as  well  as  others, 
requiring  special  methods  of  investigation  applicable  only  to 
such  minerals  as  are  transparent.  These  include  the  deter- 
mination of  the  quality,  and  numerical  values,  of  the  re- 
fraction constants  in  particular  directions,  the  so-called  axes 
of  optical  elasticity,  and  the  relation  of  these  axes  to  the 
principal  crystallographic  lines,  and  the  phenomena  of 
pleochroism.  Now,  although  the  practical  application  of 
these  methods  is,  as  a  rule,  beyond  the  power  of  beginners, 
as  they  involve  the  use  of  exact  and  somewhat  expensive 
apparatus  and  carefully  prepared  crystallographic  material, 
the  results  obtained  are  in  many  cases  of  such  great  interest 
and  importance,  especially  in  the  determination  of  crystallo- 
graphic characters  from  fragments  of  minerals  in  forms  not 
otherwise  definable,  that  a  brief  indication  of  their  character 


CHAP.  XII.]  Wave  Motion.  219 

and  the  reasoning  upon  which  they  are  founded  may  not  be 
out  of  place  here.  The  reader  is  referred  to  the  treatise  on 
'  Physical  Optics,'  by  Mr.  Glazebrook,  for  the  more  com- 
plete discussion  of  the  subject. 

In  order  to  understand  how  light  may  be  used  as  an  in- 
dicator of  the  structure  of  minerals,  it  will  be  necessary  to 
consider  as  shortly  as  possible  certain  common  optical  terms, 
and  first  of  all  the  nature  of  light  itself.  This,  according  to 
the  undulatory  theory  of  Huyghens,  which  forms  the  basis 
of  all  modern  optics,  is  a  consequence  of  the  vibratory  move- 
ment of  molecules  in  an  intangible  fluid,  or  ether,  which 
is  assumed  to  fill  the  intermolecular  space  in  all  material 
substances — solid,  liquid,  or  gaseous  alike  ;  and  by  its  con- 
tinuity forms  a  medium  for  the  transmission  of  such  move- 
ments from  their  point  of  origin,  the  luminous  source,  to  the 
optic  nerve,  where  they  become  apparent  as  light,  the  cha- 
racter of  the  movements  being  essentially  similar  to  those  of 
waves  traversing  air,  water,  or  any  other  substance.  If  a 
system  of  molecules  in  stable  equili-  FIG 

brium  be  supposed  to  be  arranged  m    a      ^       c      a     e 
line  at  regular  distances  apart,  as  in    7  /\~^  *       * 
fig-  3T9>  a  straight  line  passing  through    •, 
them  will  include  all  their  positions  of 
repose.     If,  however,  one  of  them  be  by  some  force  dis- 
placed sideways,  as  for  example  a  to  a',  the  tendency  of 
the  attractive  force  of  the  adjacent  particle  b  will  be  to  bring 
it  back  to  the  original  position,  and  it  will  after  a  time  return, 
arriving  at  a  with  sufficient  impetus  to  carry  it  on  to  a  point 
at  the  same  distance  as  a'  on  the  other  side  of  the  line  of 
equilibrium,  and  so  on  backwards  and  forwards,  a  periodical 
vibratory  motion,  analogous  to  that  of  a  pendulum,  being  set 
up  in  a  plane  determined  by  the  direction  of  the  moving 
force.     When  the  particle  a  is  displaced  it  will   exert  a 
disturbing  influence  over  the  adjacent   one  b,  tending  to 
draw  it  to  itself  as  indicated  by  the  diagonal  arrow,  which 
influence  will  be  opposed  by  the  attraction  of  c  acting  along 


22O  Systematic  Mineralogy.          [CHAP.  XII. 

the  horizontal  arrow.  The  movement  of  b  will,  there- 
fore, be  in  some  intermediate  direction,  such  as  that  of  the 
vertical  arrow,  which  it  will  swing  parallel  to  a,  and  so 
on  for  any  molecule  in  the  train  ;  the  commencement  of 
the  movements  being  progressive,  each  molecule  starting 
when  the  preceding  one  has  traversed  a  certain  dis- 

FIG.  320. 


m 

f*       t          * 

•'  r     4^^ 
1  *   .  t  i 

^^.ff 

.5 
.*» 

Vn^   f          l^^-' 
A^"2*  -?--'w  W 

tance  on  its  journey.  If  the  particle  a  in  fig.  320  be 
supposed  to  have  just  completed  one  vibration,  having 
successively  occupied  all  the  positions  indicated  by  the 
points  o  to  1 1  in  the  order  shown  by  the  arrows,  the  mole- 
cule b,  which  started  when  a  wa§J  at  i,  will  at  the  same 
Mpment  be  at  ii,  moving  downwards  towards  the  centre 
lije;  c  will  be  "a  little  farther  off  at  10,  but  moving  in 
tha  same  way;  d  will  be  at  the  greatest  distance  above 
the  line  at  9  ;  the  points  next  following,  e  and  /  will  be 
travelling  upwards,  g  will  be  passing  through  the  central 
position,  and  so  on  to  o,  which  will  be  just  about  to  com- 
mence its  first  movement.  The  points  a  to  o  therefore 
represent  the  positions  of  a  contiguous  series  of  vibrating 
molecules,  supposing  their  movement  to  be  stopped  simul- 
taneously, and  together  they  represent  the  successive  posi- 
tions passed  through  by  each  one  or  the  phases  in  its  move- 
ments :  the  line  joining  them  will  be  a  wave  line,  having  a 
crest  at  d,  a  hollow  at  /,  and  nodal  points  at  a,  g,  and  o, 
^the  horizontal  line  passing  through  them  being  the  line  of 
propagation  of  the  wave.  The  total  distance  from  a  to  o  is 
callacl  a  wave-length  ;  ag  and^^  are  half,  and  ad,  dg,  g /, 


CHAP.  XII.]  Wave  Motion.  22  r 

and  lo  quarter  wavelengths,  quantities   that  are  usually 

represented  by  the  symbols  X,  ^    and  -.       Points  in  the 

2  4 

series  which  are  half  a  wave-length  or  whose  difference  of 
phase  =  -  are  vibrating  in  opposite  directions,  as  shown  by 

2 

the  arrows  in  fig.  320. 

The  surface  containing  the  paths  of  the  molecules  is 
termed  the  plane  of  vibration,  and  the  direction  in  which  the 
wave  movement  progresses,  the  direction  of  propagation. 
In  the  simple  case  represented  these  are  at  right  angles  to 
each  other,  or  the  wave  is  a  transverse  rectilinear  wave. 

The  distance  03-09,  termed  the  amplitude  of  vibration, 
determines  the  intensity  or  brilliancy  of  the  light,  and,  as  in 
the  case  of  the  pendulum,  the  time  of  vibration  is  constant, 
whether  this  distance  be  long  or  short,  the  movement  of  the 
particle  when  at  its  maximum  speed  being  quicker  in  the 
former  than  in  the  latter  case,  but  the  total  duration  of  one 
vibrati'on  is  the  same  in  either.  The  wave-length,  on  the 
other  hand,  being  the  (^stance  travelled  during  the  time  of 
a  single  oscillation,  will  'depend  upon  the  period  of  the  latter 
being  longer  or  shorter,  according  as  the  motion  is  slower  or 
faster.  Upon  these  differences  depends  the  colours  of  the 
light  produced;  that  having  the  slowest  period  or  greatest 
length  of  wave  is  called  red,  and  that  of  the  most  rapid 
movement  and  shortest  wave-length  is  violet,  those  of  the 
colours  orange,  yellow,  green  and  blue  being  of  intermediate 
velocities  and  wave-lengths.  When  a  luminous  source  pro-  • 
duces  waves  of  only  one  length,  the  light  is  said  to  be  homo- 
geneous or  monochromatic,  but  when  waves  of  different 
lengths  are  originated  simultaneously  the  light  is  hetero- 
geneous, mixed,  or  white.  The  latter  is  the  character  of 
ordinary  sunlight,  monochromatic  light  being  characteristic 
of  the  vapours  of  elementary  bodies  when  intensely  heated 
—sodium  producing  yellow,  thallium  green,  lithium  red,  &c. 
The  light  of  candle,  oil,  or  gas  flames  is  also  mixed,  but 


222  Systematic  Mineralogy.  [CHAP.  XII. 

differs  from  sunlight  in  the  relative  proportions  of  the  con- 
stituent colours,  containing  less  blue  and  more  yellow,  some 
portion  of  the  latter  coloured  light  being  of  a  kind  never 
seen  in  sunlight. 

When  a  line  of  molecules  vibrating  in  any  direction  is 
subjected  to  a  new  impulse  acting  in  some  other  direction, 
its  movements  will  be  resolved  according  to  the  parallelo- 
gram of  forces,  and  a  new  wave  will  be  set  up,  compounded 
of  both.  In  such  cases  the  two  waves  are  said  to  interfere, 
and  the  new  one  to  be  an  interference  wave.  The  effect 
produced  by  such  interference  may  vary  very  considerably 
in  character,  according  to  the  relation  of  the  constituent 
waves.  In  the  simplest  case,  where  both  are  moving  in"  the 
same  direction,  their  planes  of  vibration  being  also  the  same, 
if  one  differs  from  the  other  by  one  or  more  complete  wave- 
lengths, the  amplitude  of  the  new  wave  will  be  augmented,  it 
being  equal  to  the  sum  of  those  of  the  original  ones. 
When  the  difference  of  phase  is  one  half  or  any  uneven 
number  of  half  wave-lengths,  the  positions  in  the  fresh  wave 
will  correspond  to  similar  ones  in&the  second,  but  their 
directions  of  motion  will  be  dissimilar.  If  therefore  the  two 
waves  be  of  different  amplitudes,  the  phases  of  that  pro- 
duced by  their  interference  will  correspond  to  those  of  the 
larger  one,  but  its  amplitude  will  only  be  equal  to  their 
difference — that  is,  the  intensity  of  the  resultant  wave  will  be 
considerably  reduced.  In  the  special  case  where  both 
waves  are  of  the  same  amplitude,  that  of  the  resultant  —  o,  or 
the  wave  motion  is  entirely  destroyed.  For  differences  of 
phase  other  than  the  above,  the  interference  waves  produced 
are  of  different  phases  and  amplitude.  Thus,  for  waves  of 
similar  amplitude  and  quarter  wave-lengths  apart,  the  aug- 
mented wave  of  interference  has  a  difference  of  phase  =  ^  X 
in  advance  of  the  first,  while  for  three-quarter  wave-length 
distance  it  is  retarded  by  £  X. 

The  wave  motions  originated  by  a  luminous  point  are 
propagated  uniformly  in  every  direction  from  it  as  a  centre, 


CHAP.  XII.]       Htiygheris  Wave  Construction. 


223 


so  that  at  any  particular  instant  a  point  in  any  one  wave 
will  be  in  exactly  the  same  condition  of  vibration  as  one  in 
any  other  at  the  same  distance  from  the  centre,  or,  in  other 
words,  points  of  similar  phase  on  the  wave  paths  will  all  be 
equidistant  from  the  centre,  and  a  surface  including  the 
whole  of  them  will  be  a  sphere  whose  radius  is  the  common 
distance.  This  surface  is  known  as  the  wave  surface  or 
wave  front,  and  the  radii  representing  the  wave  paths  as 
light  rays  ;  any  small  portion  of  the  surface  and  the  rays 
determining  it  constitute  a  beam  of  light. 

As  the  points  included  in  a  spherical  wave  surface  all 
commence  their  vibrations  simultaneously,  they  will  affect 

FIG.  32  T. 


the  equilibrium  of  the  particles  beyond  them  in  a  similar 
manner,  so  that  each  may  be  considered  as  originating  a 
particular  wave.  In  fig.  321  K  K'  represent  a  portion  of  the 
surface  of  a  wave  originating  at  c,  and  P  P'  a  similar  portion 
at  a  later  instant.  The  point  B  in  the  second  will  be  reached 
by  movements  coming  not  only  from  A  but  from  every  other 
point  of  the  surface  K  K'.  To  determine  the  effect  of  such 


224  Systematic  Mineralogy.  [CHAP.  XII. 

movements,  suppose  a  series  of  circles  LL',  PP',  R  R',  &c.,  to 
be  described  upon  the  surface  KK'  like  the  parallels  of 
latitude  on  a  globe  about  c  A  B  as  a  polar  axis  ;  all  points 
in  the  circumference  of  any  one  of  these,  such  as  P  and  P', 
will  be  equidistant  from  B,  but  at  a  different  distance  from 
those  in  any  other  circle.  The  waves  proceeding  from 
different  circles  will  therefore  have  different  distances  to 
travel,  and  as  their  period  and  rate  of  progression  are  the 
same,  they  must  arrive  at  B  in  different  conditions  of  vibra- 
tion. If  further  we  suppose  for  every  one  of  the  circles 
another  to  be  constructed  of  exactly  half  a  wave-length  greater 
distance  from  B,  the  undulations  proceeding  from  each  of 
them  will  by  their  interference  be  extinguished,  and  there- 
fore have  no  effect  at  that  point.  Such  circles  can  be  de- 
scribed for  every  portion  on  the  surface  except  A,  and 
therefore  the  whole  of  the  waves  proceeding  towards  B  will 
be  extinguished  except  those  on  the  line  A  B,  and  similarly 
only  those  on  the  prolongation  of  the  line  c  P  and  c  P'  will 
reach/  and/'  respectively.  From  this  it  follows  that  although 
every  point  on  the  wave  surface  is  the  centre  of  a  new  wave, 
the  motion  of  the  latter  will  only  be  apparent  at  the  point 
where  it  touches  the  surface  forming  a  common  envelope  to 
the  whole  of  the  waves  of  the  same  class.  This  is,  in  the 
case  in  question,  the  spherical  surface  //',  or  the  wave  front 
corresponding  to  the  later  period.  Upon  this  principle  is 
founded  Huyghens'  wave  construction,  whereby  the  wave 
front  at  any  given  moment  may  be  found  if  the  originating 
surface  and  the  rate  of  propagation  be  given.  This  is 
done  by  describing  upon  points  of  origin,  such  as  A  p  and  P' 
semicircles  whose  radii  are  proportional  to  the  velocity  of 
wave  transmission  at  those  points,  and  drawing  through  the 
points  /  B/'  the  common  enveloping  surface,  which  is  the 
second  wave  front  as  before. 

If  the  point  of  origin  of  the  wave  motion  be  situated  at  a 
great  distance,  the  rays  contained  in  a  beam  of  light  may  be 
considered  as  sensibly  parallel,  and  the  included  portion  of 


CHAI-.  XII.]  Wave  Motion.  225 

the  wave  front  as  a  plane  surface.  This  is  termed  a  plane 
wave.  Ordinary  solar  or  daylight  is  of  this  character,  the 
distance  of  the  luminous  centre,  the  sun,  being  so  great  that 
a  beam  of  light  must  have  a  diameter  of  nearly  a  thousand 
miles  for  the  radii  limiting  it  to  diverge  by  an  angle  of  one 
second  from  parallelism,  so  that  in  most  instances  of  natural 
illumination  the  beam  may  be  considered  as  a  cylinder  made 
up  of  parallel  rays. 

The  power  of  transmitting  wave  movements  uniformly 
in  one  or  more  directions  supposes  the  medium  possessing 
it  to  be  homogeneous  or  uniform  in  molecular  structure.  If 
the  structure  is  such  that  the  rate  of  transmission  is  exactly 
the  same  in  any  direction,  the  medium  is  said  to  be  isotropic. 
This  represents  the  most  complete  and  symmetrical  kind  of 
homogeneity. 

In  the  second  kind  of  homogeneous  media,  known 
as  anisotropic  or  heterotropic,  the  rate  of  progression  of  wave 
movements  varies  with  their  direction,  but  is  constant  for 
any  similar  direction.  In  heterogeneous  or  non-homogeneous 
media,  on  the  other  hand,  there  is  no  relation  between 
direction  and  velocity  of  wave  propagation,  the  latter  being 
subject  to  variation  even  when  the  former  is  constant.  The 
movements  in  such  a  medium  can  only  be  reduced  to  order 
by  supposing  them  to  be  made  up  of  portions  of  homo- 
geneous substances  of  different  properties,  and  treating  each 
one  separately. 

The  velocity  of  propagation  of  wave  movements  changes 
when  they  pass  from  one  medium  into  another,  the  different 
velocities  being  related  to  each  other  according  to  a  funda- 
mental proposition  in  mechanics,  directly  as  the  square 
roots  of  their  coefficients  of  elasticity  e,  and  inversely  as 
the  square  roots  of  their  densities  ;  or  if  c  and  c1  represent 
the  different  velocities,  their  ratio  will  be  expressed  by 


226  Systematic  Mineralogy.          [CHAP.  XIL 

When  e  is  nearly  of  the  same  value  in  the  two  media  the 
velocity  will  diminish  as  d  increases,  or  generally  the  velo- 
city of  propagation  is  less  in  the  denser  medium. 

In  passing  from  one  medium  to  another  of  different 
density,  the  entire  wave  movement  is  never  completely 
transferred.  If  the  second  medium  has  the  lesser  density, 
the  energy  of  a  vibrating  molecule  at  the  limiting  surface  of 
the  first  will  be  more  than  sufficient  to  establish  an  equality 
of  movement  in  a  similar  molecule  in  the  second,  and  there- 
fore part  of  the  original  motion  will  be  returned  in  the  form 
of  a  backward  wave,  whose  phase  at  any  point  will  be 
exactly  the  same  as  that  of  the  original  wave  would  have 
been  at  a  point  similarly  situated  in  front. 

On  the  other  hand,  if  the  second  medium  be  the  denser, 
not  only  will  the  original  movement  be  entirely  absorbed, 
but  a  return  movement  will  be  set  up  in  the  molecules  of 
the  first  in  the  form  of  a  wave  of  exactly  opposite  phase  to 
the  original  one.  In  either  case  the  change  is  accompanied 
with  the  same  result — a  portion  of  the  original  movement 
only  enters  or  is  transmitted  by  the  second  medium,  the 
remainder  being  diverted  or  reflected  at  the  surface  of  con- 
tact. This  proposition  holds  good  in  all  cases.  Whenever 
light  passes  from  one  medium  to  another,  a  portion  is  in- 
variably lost  by  reflection,  and  the  intensity  of  the  trans- 
mitted beam  is  diminished,  the  loss  increasing  with  the 
difference  in  density  of  the  two  media.  Where  the  difference 
is  so  great  that  none  or  very  little  of  the  light  enters,  the 
denser  substance  is  said  to  be  opaque  ;  but  if  all,  or  nearly 
all,  is  transmitted,  it  is  transparent  or  translucent.  No 
substance  is  known  to  be  either  completely  opaque  or  per- 
fectly transparent,  as  the  light  reflected  by  the  most  charac- 
teristic examples  of  the  former,  properly  polished  metal 
surfaces,  is  always  modified  to  some  extent,  showing  that 
penetration  has  taken  place,  though  only  in  a  very  small 
degree  ;  and  those  that  can  be  reduced  to  a  sufficiently  thin 
laminae  such  as  gold  and  silver,  are  actually  found  to  be 


CHAP.  XII.] 


Reflected  Wave, 


227 


translucent,  and  the  most  limpid  translucent  substances  in 
sufficiently  thick  masses  sensibly  diminish  the  light  passing 
through  them.  The  light  so  lost  is  said  to  be  absorbed. 
When  the  absorption  affects  waves  of  all  lengths  similarly, 
the  result  is  only  a  diminution  in  the  brilliancy  of  the  light ; 
but  selective  or  unequal  absorption  of  particular  kinds  of 
waves  alter  the  character  of  the  transmitted  light,  giving 
rise  to  the  phenomena  of  colour. 

The  nature  of  the  wave  surface  produced  by  reflection 
may  be  found  by  the  construction  given  in  fig.  322.     If 


MN  in  fig.  322  represent  the  surface  of  contact  of  two  dis- 
similar isotropic  media,  and  A  the  point  of  origin  of  wave- 
motion  in  the  first,  the  arc  B  B'  will  be  the  section  of  the 
spherical  wave-front  at  the  moment  when  the  ray  A00 
meets  M  N  ;  the  more  divergent  rays  A0L  to  A#4  on  either 
side  arriving  at  the  same  surface  progressively  at  later 
periods.  At  each  of  these  points  a  reflex  wave-movement 

Q  2 


228 


Systematic  Mineralogy.          [CHAP.  XII. 


originates,  that  from  a  0,  being  the  earliest,  arrives  at  the 
point  e0  at  the  same  moment  that  those  from  ala2a3  and  a4 
reach  el  e2  <?3  and  e4  respectively,  and  the  outer  rays  A  M  and 
AN  have  just  arrived  at  the  surface  MN.  If,  therefore, 
semicircles  be  described  from  a0  with  the  radius  aQ  e0,  from 
at  with  #!  <?u  from  az  with  a2  e%,  etc.,  the  curve  forming 
their  common  tangent,  the  circular  arc  M  e0  N,  whose  radius 
is  c  a0  +  a0  eQ,  will  be  the  wave-front  required.  If  an  arc 
be  struck  with  the  same  radius  from  A  passing  through 
MN,  it  will  represent  the  position  that  the  original  wave- 
front  would  have  reached  had  the  motion  been  continued 
in  the  first  medium.  Hence  we  see  that  the  direct  and 
reflected  wave-surfaces  are  exactly  similar  in  dimension, 
but  reversed  in  position. 

The  direction  of  reflection  in  the  central  ray  A  a0  is  the 
same  as  that  of  its  arrival  or  incidence  —  i.e.,  normal  to  the 
reflecting  surface  ;  but  in  the  rays  on  either  side  these  direc- 
tions differ,  making  longer  angles  with  each  other  as  the 
obliquity  of  incidence  increases.  If  through  each  of  the 
reflecting  points,  a}  to  a4,  a  line  be  drawn  normal  to  M  N,  it 
will  be  symmetrical  to  the  two  rays  at  that  point,  or  they 


FIG.  323. 


will  make  equal  angles  with 
it  and  consequently  with  the 
reflecting  surface.  This  is  ex- 
pressed in  the  following  pro- 
position :  The  angle  of  inci- 
dence is  equal  to  the  angle  of 
reflection. 

The  direction  of  rays  pass- 
ing into  the  second  medium 
is  found  by  the  construction 
shown  in  fig.  323.  If  A,  A2  A3  A4 
represent  rays  coming  from  a 
luminous  source  at  a  great 
distance,  and  therefore  parallel,  the  surface  represented  by 
the  line  al  B  normal  to  them  will  be  the  plane  wave-  front  at 


CHAP.  XII.] 


Refraction. 


229 


the  moment  of  incidence  of  the  ray  AI  a^  with  the  surface 
limiting  the  two  media  M  N.  Supposing  the  density  of  the 
second  medium  to  be  such  that  the  velocity  of  wave  propa- 
gation in  it  is  only  one-half  of  that  in  the  first,  the  semi- 
circles described  about  a^  a^  and  a3,  with  radii  corresponding 
to  half  the  distances  EI  N,  B2N,  and  BSN  respectively  will 
represent  the  positions  of  waves  set  up  at  these  points  at 
the  moment  when  the  most  distant  ray,  A4,  arrives  at  the 
point  M,  and,  according  to  the  principles  previously  laid 
down,  the  plane  represented  by  the  line  NRI}  which  is 
tangent  to  all  of  them,  will  be  the  wave-front  of  the  whole 
beam.  As  this  is  also  a  plane  wave,  the  parallelism  of  the 
rays  is  not  altered,  but  their  direction,  indicated  by  the 
lines  normal  to  R!  N  passing  through  al  a%  and  a3  is  changed, 
the  individual  rays  being  bent  or  refracted  towards  a 
normal  to  the  plane  of  incidence  passing  through  the  same 
points.  If,  on  the  other  hand,  the 
first  be  the  denser  medium,  the 
velocity  of  propagation  will  be 
doubled  in  the  second,  and  the 
rays,  as  shown  in  fig.  324,  will  be 
bent  away  from  the  normals  to 
the  plane  of  incidence,  or  the 
refractive  angles  will  be  in- 
creased. The  amount  of  the 
refraction  is  measured  in  fig.  325 
for  the  rays  A,  o  and  o  R,  by  the 
lines  A!  n\  and  Rt  n3,  which  are 

to  each  other  the  sines  of  their  respective  angles  with  the 
vertical  line  nop,  or  calling  the  larger  angle  /  and  the 
smaller  one  r, 


FIG.  324. 


sm. 


sn.  r 


-  =  n. 


This   quantity,  which   is  constant  in  an  isotropic  sub- 
stance  for  any  angle   of  incidence,   and   corresponds  to 


230 


Systematic  Mineralogy.          [CHAP.  XII. 


FIG.  325. 


the  ratio  of  the  respective  velocities  of  wave  transmission 

or  -- ,  is  termed  the  qiiotient,  exponent,  or  index  of  refraction , 

v\ 

the  last  or  the  equivalent  term,  refractive  index,  being  most 
generally  used  The  value  of  n  varies  for  the  greater  number 
of  transparent  isotropic  minerals  between  i  -3  and  1 7,  that 
of  air  being  taken  as  unity,  the  maximum  of  2^4  being 
attained  in  Diamond.  A  few  anisotropic  substances  such 
as  the  Ruby,  Silver  Ores,  and  Cinnabar,  afford  examples  of 
still  higher  refractive  indices,  which  reach  or  even  slightly 
exceed  3*0. 

A  more  oblique  ray,  such  as  A20  in  fig.  325,  whose  angle 
of  incidence  is  *',  is  refracted  in  the  denser  medium  in 

the  direction  o  R2,  or  has  the 
larger  angle  of  refraction  r1, 
but  the  ratio  of  the  quotient  of 
their  sines  is  not  altered — i.e., 

sin.  i'        sin.  / 

—. -.  =  —. —  =  n 

sin.  r         sin.  r 

as  before  ;  or  the  deviation  of 

P. ~-  the  refracted  ray  varies  with 

the  angle  of  incidence.  The 
maximum  obliquity  of  inci- 
dence is  in  the  direction  M  <?, 
or  parallel  to  the  dividing 
surface,  in  which  it  is  clear 
there  can  be  no  refraction,  as 
the  path  of  the  ray  is  entirely 
Also,  if  the  ray  meet  the 
plane  of  incidence  at  right  angles,  or  z  =  o°,  the  deviation 
of  the  refracted  ray  r  also  =  o°,  or  there  is  no  deviation, 
and  the  ray  follows  the  same  direction  in  both  media 
without  refraction.  For  any  other  direction  sin.  /  is 


within   the    first    medium. 


always  greater  than  sin.  r,  as  sin.  r  = 


sin,  t 
n 


If  the  first 


medium  be  denser  than  the  second,  the  velocity  of  trans- 


CHAP.  XII.] 


Total  Reflection. 


231 


mission  will  be  greater  in  the  latter  than  the  former,  the 
wave-front  of  the  transmitted  beam  R,  N,  fig.  326,  will  make 
a  longer  angle  with  M  N  than  that  of  the  incident  one  M  BJ, 
the  angles  of  refraction  of  the  individual  rays  will  be  larger 
than  their  angles  of  incidence,  and  will  increase  with  the 
deviation  of  the  latter  from  the  vertical,  but  more  rapidly, 
as  will  be  apparent  by  turning  fig.  325  upside  down  and 
considering  the  positions  of  the  refracted  and  incident  rays 
to  be  transposed.  In  this  case  sin.  r  =  m  sin.  /,  and  as 

sin.  90°  =  i  when  sin.  /  =  — ,  r  =  9o°,\pr  the  refracted  ray 


he  construction 
326. 


will  coincide  with  the  plane  of  separation, 
for  the  special  case  n  =  2  and 
r  =  90°  is  seen  in  fig.  326,  where 
the  vertical  line  through  N  is  the  A 
tangent  plane  or  wave-front  of  the 
refracted  beam,  and  M  n  the  direc- 
tion of  the  refracted  ray.  The 
radius  of  the  circle  M  K  =  2  BJ  N, 
and  A,  n  =  sin.  /=  \  or  /=  30°. 
For  this  particular  value  of  the 
refractive  index,  therefore,  all  rays 
whose  inclination  to  the  normal 
to  the  plane  of  incidence  is  thirty 
degrees  or  above  are  incapable 
of  passing  into  the  less  dense  medium,  but  are  reflected  at 
the  surface  of  separation.  This  property  is  known  as  total 
reflection,  and  the  minimum  angle  at  which  it  takes  place 
is  the  limiting  angle  of  refraction,  or  the  angle  of  total  re- 
flection. 

When  the  denser  medium  is  limited  by  parallel  surfaces, 
the  angle  of  incidence  of  the  refracted  ray  at  the  second 
surface  will  be  the  same  as  that  of  its  first  refraction,  or 
/'  =  r  (fig.  327),  and  therefore  its  angle  of  refraction,  r1^  on 
emergence  into  the  first  medium,  will  be  the  same  as  that  of 
original  incidence,  /,  or,  in  other  words,  the  ray  will  resume 


232 


Systematic  Mineralogy.          [CHAP.  xn. 


its  original  direction.  A  ray  of  light  is  not  therefore  altered 
in  direction  by  passing  through  any  transparent  substance 
bounded  by  parallel  surfaces.  If,  however,  the  surfaces 
are  not  parallel,  the  ray  at  emergence  will  be  more  or 
less  changed  from  its  original  direction,  supposing  the  in- 
clination of  the  surfaces  not  to  be  sufficient  to  produce 
total  reflection.  The  amount  of  this  deviation  in  direction 
may  be  calculated  when  the  inclination  of  the  limiting  sur- 
faces and  the  refractive  index  of  the  medium  are  known, 


FIG.  327. 


FIG.  328. 


EZS 


P' 


and  conversely  from  the  measured  deviation  produced  by 
a  refracting  body  or  prism  of  known  angle,  the  refractive 
index  of  the  substance  composing  it  may  be  found.  The 
principle  of  this  method  is  shown  in  fig.  328,  where  PP'P" 
is  the  section  of  a  prism  of  the  refractive  angle  o  in  a  plane 
perpendicular  to  its  refracting  edge  P.  The  ray  A  M  incident 
at  the  angle  /  on  P  P'  is  refracted  at  the  angle  r  towards 
M',  where  it  becomes  incident  at  the  angle  i'  upon  P  P",  is 
refracted  at  the  angle  r1,  and  follows  the  direction  M'R 


CHAP.  XII.]  Minimum  Deviation,  233 

after  emergence.  If  the  line  M'  A'  be  drawn  parallel  to 
A  M,  the  angle  A'  M  R  =  I  will  measure  the  total  deviation 
of  the  ray  from  its  original  course  effected  by  the  prism. 
Next,  from  the  point  M'  draw  the  lines  M'  A'  and  M'  T,  the 
first  parallel  to  the  original  direction  of  the  ray  A  M,  and 
the  second  to  the  normal  n  M,  produce  the  common  path 
of  the  first  refracted  and  second  incident  ray  M  M'  towards 
K,  and  continue  the  normals  «M  nfu'  to  their  point  of 
intersection  at  s.  The  angles  formed  at  M'  will  have  the 
following  signification : 

n'  M'  K  =  i1  =  K  M'  T  =  r,  A'  M'  T  =  n  M  A  =  /,  n'  M'  R  =  r1, 

and  A'  M'  R  =  $,  or  the  deflection  of  the  ray  from  its  original 
course  by  the  prism.  But 

A'  M'  R  =  <)  —  A'  M'  K  +  K  M'  R  =  (T  M'  A'  —  T  M'  K)  + 
(R  M'  n'  —  n'  M'  K) 

or  a  =  (/  -  r)  +  (r'  -  i1}  =  i  +  r1  -  (r  +  i'). 

In  the  triangle  PM  M'  a  +  (90°  —  r)  +  (90°  —  f)  =  180°, 
consequently,  a  =  r+  i1  and  S  =  /  +  r1 —a, 

Fig.  328  is  so  constructed  that  i=r'  and  r  =  tl,  for 
which  case  S  =  2  i  —  a  =  2  r'  —  a,  and  has  a  minimum 
value. 

If  one  angle  is  greater  than  the  other,  e.g.  ift'—r'  +  ft,  or 
r1  —  i  +  (3,  the  deviation  will  be  in  the  first  case  (J  =  2  r'  + 
/3  —  a,  and  in  the  second  I  =  2  i  +  ft  —  a,  either  of  which 
is  obviously  greater  than  2  i  —  a.  This  latter,  therefore,  is 
the  minimum  angle  of  deviation,  and  is  produced  when  the 
ray  makes  equal  angles  with  the  faces  of  the  prism  both  at 
incidence  and  emergence.  Further,  as 

a  =  2  /  —  a,  /  = ; 

2 

and  as 

r  =  /',  u=  r  +  /'  =  2  r,  r  =  -, 

2 


234  Systematic  Mineralogy.          [CHAP.  XII. 

and 


a 

sm. 
_  sm.  a 

~ 


sn'r          sin." 

2 

or  the  refractive  index  of  the  substance,  if  isotropic,  is  equal 
to  the  sine  of  half  the  sum  of  the  refractive  angle  of  the 
prism  and  the  angle  of  minimum  deviation  divided  by  the 
sine  of  half  the  angle  of  minimum  deviation.  This  is  the 
simplest  and  most  direct  method  of  determining  refractive 
indices,  as  two  faces  of  a  crystal  may  be  used  as  a  prism 
if  their  angle  is  not  too  large  ;  from  40°  to  70°  are  the 
most  convenient  angles  when  the  refractive  indices  are 
moderately  high  (  i  '5  to  i'7).  In  all  cases  the  determina- 
tion must  be  made  for  particular  parts  of  the  spectrum,  either 
by  using  monochromatic  light  or  by  observing  the  deviation 
of  'the  principal  dark  lines  in  the  solar  spectrum.  For  the 
former  purpose  the  flame  of  a  Bunsen's  burner  coloured  red 
by  a  bead  of  sulphate  of  lithium  ignited  at  the  end  of 
a  platinum  wire,  yellow  by  chloride  of  sodium,  or  green 
by  sulphate  of  thallium,  is  the  most  convenient  kind  of 
illumination.  The  angle  a  is  measured  with  a  reflecting 
goniometer  and  the  deviation  by  the  same  instrument 
arranged  to  allow  the  prism  and  telescope  to  move  inde- 
pendently. 

Another  method  of  determining  refractive  indices  origi- 
nally proposed  by  the  Due  de  Chaulnes  in  1767,  and  re- 
cently systematised  and  extended  by  Sorby  and  Stokes,1 
consists  in  measuring  the  displacement  of  the  focal  point  of 
a  lens  or  compound  microscope  adjusted  to  a  distinct  vision 
of  an  object  in  air  when  a  parallel  plate  of  known  thickness 
of  a  denser  substance  is  interposed  between  the  object  and 
the  lens.  The  effect  of  this  interposition  is  to  bring  the 

1  Proc.  Royal  Soc.  xxvi.  p.  384.  Journal  of  the  Royal  Micro- 
scopical Society.,  1878. 


CHAP.  XII.]  Single  Refraction.  235 

apparent  place  of  the  image  nearer  to  the  lens,  so  that  the 
latter  has  to  draw  back  through  a  small  distance,  d,  whose 
amount  depends  upon  the  thickness  of  the  plate  /,  and  its 
refractive  index  n,  if  isotropic,  the  relation  of  these  quantities 

being  expressed  by  the  formula,  n  =  -. 

l>   ^~  & 

This  method  has  the  advantage  of  being  applicable  to 
very  thin  and  minute  crystals  or  parallel  plates,  and  is  there- 
fore useful  in  the  microscopic  study  of  rocks  and  minerals, 
but  the  measuring  apparatus,  which  may  be  either  a  vernier 
scale  or  micrometer  screw,  but  preferably  the  latter,  must  be 
made  with  great  accuracy,  as  the  quantities  to  be  measured 
are  usually  very  small. 

In  an  isotropic  substance  the  velocity  of  propagation  of 
wave  motion  being  alike  in  all  directions,  the  refractive 
index  will  also  be  constant  in  any  direction  for  light  of 
any  particular  colour,  being  lowest  for  the  red  and  highest 
for  the  blue  end  of  the  spectrum.  Such  substances  are 
therefore  said  to  be  single-refracting  or  monorefringent 
They  include  all  transparent  homogeneous  gases  and  liquids, 
the  latter  with  a  few  exceptions ;  and  among  solids,  both 
those  that  are  amorphous  and  those  crystallising  in  the 
cubic  system,  all  other  crystallised  substances  being  iniso- 
tropic. 

The  properties  of  a  ray  of  ordinary  light  are  similar  in 
any  radial  direction  about  its  line  of  transmission  considered 
as  an  axis ;  it  appears  to  vibrate  simultaneously  in  these 
directions.  There  is,  however,  good  reason  to  suppose  that 
fhis  is  not  actually  the  case,  and  that  the  directions  or 
azimuths  of  vibration  change  continuously,  but  so  very 
rapidly  that  the  changes  are  not  perceptible  to  the  eye. 
Upon  this  supposition  the  plane  of  vibration  of  a  molecule, 
<?,  fig.  329,  would  be  changed  by  a  small  anglfc  at  each  pul- 
sation, or  if  the  limit  of  its  first  vibration  is  //,  on  the  line 
o  N,  that  of  the  second  will  be  somewhere  to  the  right  of  N 
(if  the  change  is  in  that  direction),  that  of  the  third,  some- 


236 


Systematic  Mineralogy.          [CHAP.  XII. 


FIG.  329. 


what  further  on  the  same  side,  and  so  on,  until  the  direction 
of  vibration  is  on  the  line  o  e.  The  motion  in  any  interme- 
diate plane,  such  as  o  c,  may  be  considered  as  the  resultant 
of  two  forces  acting  simultaneously,  on  towards  N,  and  oe 
towards  E  ;  and  if  the  velocity  be  the  same  in  any  direction 

the  amplitude  of  the  vibration  in 
oc  will  be  exactly  the  same  as 
those  in  the  NS  or  EW  planes, 
and  consequently  the  intensity  of 
the  light  will  be  unchanged,  what- 
ever may  be  the  azimuth  of  its 
plane  of  vibration.  These  rela- 
tions, however  only  hold  good 
for  isotropic  media,  as  in  ani- 
sotropic  ones  the  velocity  of  wave 
propagation  varies  with  the  direc- 
tion, the  differences  being  greatest  between  directions  at 
right  angles  to  each  other.  Supposing  NS  to  be  such  a 
direction  of  maximum  velocity,  E  w  will  be  that  of  the  corre- 
sponding minimum,  and  the  molecule  o  will  vibrate  in  either 
of  these  with  its  full  proper  intensity ;  but  when  in  the 
direction  o  c  the  intensity  will  not,  as  in  the  previous  instance, 
be  {hat  due  to  the  components  on,  oe,  as  these  impulses 
having  different  velocities  cannot  arrive  simultaneously  at 
c,  but  each  will  act  independently,  and  consequently  the  ray 
will  be  resolved  into  two,  whose  intensities  will  vary  with 
their  azimuths,  one  of  them  having  its  greatest  intensity  in 
the  direction  'o  N,  diminishing  to  o  n  in  the  direction  o  c,  and 
to  nothing  in  o  E,  or  in  proportion  to  the  cosines  of  the  azi- 
muths, and  the  other  having  its  maximum  in  o  E  and  dimin- 
ishing to  nothing  in  o  N  or  o  s,  or  as  the  sines  of  the  azimuth ; 
but  as  these  changes  take  place  too  rapidly  to  be  followed 
by  the  eye,  trie  resulting  effect  is  the  production  of  two  rays, 
each  of  half  the  intensity  of  the  original  ray  of  ordinary  light, 
and  vibrating  in  one  plane  which  is  at  right  angles  to  the 
corresponding  plane  of  the  other  ray.  Such  rays  are  said 


CHAP.  XII.] 


Double  Refraction. 


237 


FIG.  330. 


to  be  polarised  in  planes  at  right  angles  to  their  planes  of 
vibration,  and  are  no  longer  freely  transmissible  in  any 
direction  like  those  of  ordinary  light,  and  as  a  consequence 
of  their  differences  in  velocity  they  will  have  different 
refractive  indices,  or  the  substances  producing  them  are 
doubly  refracting  or  birefringent. 

The  general  phenomena  observed  in  a  doubly  refracting 
crystal  are  best  seen  in  Calcite,  in  which  the  property  is 
very  strongly  developed.1  If  abed  (fig.  330)  be  the  section 
of  a  cleavage  rhombohe- 
dron  so  placed  that  a  b  and 
c  d  are  the  shorter  diagonals 
of  a  pair  of  faces,  a  c  and 
bd  polar  edges,  and  ad  the 
principal  axis,  a  ray  of  light, 
r,  incident  at  n  in  a  direc- 
tion normal  to  a  b,  will  be 
divided  into  two  parts  ;  one 
following  the  law  of  ordi- 
nary refraction  will  pass 
through  unchanged  towards 

<?,  while  the  other  will  be  refracted,  and  assuming  the 
original  direction  on  emergence  at  /  will  travel  in  a  direc- 
tion parallel  to  the  first,  but  at  some  distance  from  it,  to- 
wards e,  so  that  by  looking  through  the  crystal  in  the 
direction  no  towards  a  brightly  illuminated  object  on  a 
dark  ground,  two  images  of  it  will  be  seen,  one  at  o  and 
the  other  at  e.  The  first  of  these  rays  is  called  the  ordinary 
and  the  second  the  extraordinary  ray.  If  the  direction 
of  incidence  be  oblique,  both  rays  will  be  refracted,  but 
in  different  degrees  :  the  ordinary  ray  following  the  law  of 
sines  and  having  a  constant  index,  w  =  i  '658  for  yellow 
light,  whatever  may  be  the  angle  of  incidence  ;  while  that 
of  the  extraordinary  one  varies  with  its  direction  in  the 

1  By  strong  double  refraction  is  meant  that  the  indices  of  the  rays 
differ  considerably,  not  that  their  absolute  values  are  very  high. 


238  Systematic  Mineralogy.         [CHAP.  xii. 

crystal,  being  at  a  minimum  of  £  =  1*486  parallel  to  c  b,  and 
at  a  maximum  which  is  the  same  as  that  of  the  ordinary  ray 
when  parallel  to  the  principal  axis  a  d.  As  the  same  rela- 
tion holds  good  for  any  of  the  three  lateral  interaxial  planes, 
which  in  common  with  all  planes  passing  through  the  ver- 
tical axis  are  called  principal  optical  sections,  it  follows  that 
both  rays  will  have  the  same  index,  or  there  will  be  only 
single  refraction  in  the  direction  of  the  vertical  axis.  If, 
therefore,  an  object  be  viewed  in  this  direction,  which  can 
be  done  if  the  polar  solid  angles  are  ground  and  polished 
into  faces  parallel  to  the  horizontal  plane,  or  a  transparent 
natural  crystal  terminated  by  the  basal  pinakoid  is  used, 
only  one  image  will  be  apparent,  but  in  every  other  direc- 
tion there  will  be  two,  their  distance  apart  being  greatest 
when  seen  at  right  angles  to  the  principal  axis.  This 
direction  of  single  refraction  is  termed  an  optic  axis,  and 
crystals  in  which  one  such  direction  is  apparent  are  said 
to  be  optically  uniaxial  or  uniaxal.  These  include  all 
those  belonging  to  systems  in  which  a  single  principal  axis 
can  be  distinguished — i.e.,  the  hexagonal  and  tetragonal 
systems. 

The  velocity  of  propagation  of  light  waves  being  inversely 
proportional  to  the  refractive  indices  of  the  medium,  that 
of  the  ordinary  ray  will  be  the  same  in  any  direction,  or  its 
wave  surface  will  be  a  sphere  ;  but  in  the  extraordinary  ray 
it  differs  according  to  the  angle  made  by  the  ray  with  the 
optic  axis,  being  the  same  as  that  of  the  ordinary  ray  when 
parallel,  and  at  a  maximum  difference  from  it  when  at  right 
angles  to  that  axis.  For  any  intermediate  direction,  such  as 
op,  fig.  331,  it  will  be  equal  to  the  radius  vector,  corre- 
sponding to  the  angle  t,  of  the  ellipse,  whose  semi-axes  are 
the  greatest  and  least  velocities,  oc,  oa.  But,  as  these  con- 
ditions hold  good  for  any  plane  containing  the  optic  axis,  it 
follows  that  the  wave  surface  of  the  extraordinary  ray  will 
be  the  spheroid  or  ellipsoid  of  revolution  generated  by  the 
rotation  about  a  o,  of  the  complete  ellipse  of  which  ape  is  a 


CHAP.  XII.]         Uniaxial  Wave  Surface.  239 

quadrant.  In  Calcite,  where  the  direction  of  minimum 
velocity  is  parallel  to  the  optic  axes,  the  spheroid  is  oblate, 
or  the  extraordinary  encloses  the 

.  FIG.  331. 

ordinary  wave  surface ;  but  in  the 
other  class  of  uniaxial  species,  of 
which  Quartz  may  be  taken  as  the 
type,  the  optic  axis  is  the  direc- 
tion of  maximum  velocity,  and 
therefore  the  extraordinary  wave 
spheroid  is  prolate,  and  enclosed 
by  the  sphere  representing  the 
ordinary  wave  surface.  These 
conditions  are  represented  in  fig. 
331,  if  oc  be  turned  into  the  vertical  position,  when  ca' 
will  be  the  section  of  the  ordinary  wave,  while  cpa  re- 
mains the  extraordinary  one.  In  the  former  class,  for  any 
doubly  refracted  ray  in  the  principal  optic  section  of  the 
crystal,  the  extraordinary  has  a  lower  refractive  index  than 
the  ordinary  ray,  arid  will  therefore  be  deviated  from  the 
original  direction  less  than  the  latter,  or  make  a  larger 
angle  with  the  optic  axis  than  the  ordinary  ray.  These  are 
called  negative  crystals.  In  the  other  or  positive  class  of 
uniaxial  crystals,  in  any  direction  involving  double  refrac- 
tion, the  higher  index  is  always  that  of  the  extraordinary 
ray,  which  will  therefore  make  a  smaller  angle  with  the 
optic  axes  than  the  ordinary  ray. 

For  the  determination  of  the  refractive  indices  of  a 
uniaxial  crystal  by  the  method  of  minimum  deviation,  a 
prism  is  required  having  a  refracting  edge  either  parallel  to 
the  optic  axis,  or  perpendicular  to  it ;  but  in  the  latter  case 
the  faces  of  the  prism  must  make  equal  angles  with  the  optic 
axis.  The  deviation  of  the  two  rays  is  observed  successively 
in  one  operation,  each  one  being  alternately  extinguished 
by  a  Nicol's  prism  interposed  in  the  path  of  the  beam  in 
the  manner  subsequently  described. 

In   the   second   great  division  of  anisotropic  crystals, 


240  Systematic  Mineralogy.          [CHAP.  XII. 

those  belonging  to  the  rhombic,  oblique,  and  triclinic 
systems,  the  optical  phenomena  are  more  complex,  the 
differences  in  the  velocities  of  wave  transmission  and  re- 
fractive indices  in  different  directions  not  being  related  to 
the  principal  crystallographic  lines  in  any  simple  manner. 
In  any  such  crystals  there  will  be  two  directions  in  which 
the  molecular  elasticity,  and,  consequently,  the  velocity  of 
wave  transmission,  will  be  respectively  the  maximum  and 
minimum  of  all  the  possible  values  ;  and  these  are  at  right 
angles  to  each  other  in  the  same  plane,  while  in  a  third 
direction,  at  right  angles  to  both,  the  value  will  be  an 
intermediate  one.  These  directions  are  known  as  the  axes 
of  maximum,  minimum,  and  mean  *  elasticity,  respectively. 
In  such  a  system,  all  three  axes  being  dissimilar,  directions 
of  equivalent  refractive  value  cannot  be  symmetrical  to  any 
one  axis,  as  in  the  uniaxia"!  class,  and  therefore  the  wave 
surface  cannot  be  a  spheroid. 

If  ox  and  oz,  fig.  332,  be  the  axes  of  maximum  and 
minimum  elasticity,  and  the  position  of  OY,  the  axis  of 
mean  elasticity,  be  imagined  as  perpendicular  to  the  plane 
of  the  page,  a  ray  of  ordinary  light,  following  the  direction 
ox,  would  vibrate  successively  in  every  possible  azimuth 
in  the  normal  plane  containing  OY  and  oz,  and,  conse- 
quently, on  entering  the  crystal,  would  be  resolved  into 
two  rays,  each  vibrating  parallel  to  one  of  these  axes  and 
moving  with  its  characteristic  velocity.  If,  therefore,  the 
motion  originate  at  o,  the  faster  moving  ray,  whose  velocity 
will  be  that  corresponding  to  the  mean  elasticity,  will  arrive 
at  B,  when  the  slower  one  has  only  reached  c,  which,  with 
corresponding  positions  on  the  opposite  side,  will  be  points 
on  the  wave  section.  Similarly,  a  ray  following  the  direc- 
tion of  minimum  elasticity,  oz,  is  resolved  into  two  vibrating 
parallel  to  ox  (maximum)  and  OY  (mean),  whose  propor- 
tional paths  are  o  A  and  o  B.  For  any  intermediate  direc- 

1  The  word  mean  is  here  used  in  the  sense  of  intermediate,  and  does 
not  refer  to  any  particular  mean. 


CHAF.  XII.]          Biaxial  Wave  Surface. 


241 


tion,  such  as  o  L,  the  paths  of  the  resolved  rays  are  o  B,  and 
as  the  velocity  of  the  slower  ray  is  greater  than  the  minimum, 
but  less  than  the  mean,  the  points  B  and  L'  are  nearer  to- 
gether than  B  and  c;  while  for  os,  which  is  nearer  the 
maximum  than  the  minimum  axis,  the  corresponding  paths 
are  o  B  (mean)  and  o  T'  (greater  than  the  mean  and  less  than 


the  maximum),  and  consequently  T'  and  B  are  nearer  together 
than  A  and  B  are.  The  point  B  being  equidistant  from  o, 
the  curve  passing  through  them  all  is  a  circle  whose  radius  is 
proportional  to  the  mean  velocity ;  while  o  c,  o  L',  o  T',  and 
OA,  are  radii  of  the  ellipse  having  the  semi-diameters  oc 
and  o  A,  whose  lengths  represent  the  corresponding  minimum 
and  maximum  velocities.  The  wave  is  therefore  bounded 
by  two  surfaces,  of  different  curvatures,  which  cross  each 
other  at  K.  This  point  being  common  to  both  curves,  the 
resolved  rays  following  o  K  would  have  the  same  velocity 
in  the  crystal;  but  as  the  tangent  planes  o 0,  xx,  repre- 

R 


242  Systematic  Mineralogy.          [CHAP.  XII. 

senting  their  plane  wave  surfaces  are  differently  inclined  to 
o  K,  owing  to  the  difference  in  the  curvature  of  the  two 
surfaces,  they  will  be  dissimilarly  refracted  at  emergence. 
The  two  surfaces  have,  however,  a  common  tangent 
plane,  whose  contact  line — a  small  circle  of  the  diameter 
uu'  (fig.  333)— is  the  base  of  a 
cone  whose  apex  is  at  o.  The 
rays,  therefore,  whose -paths  in  the 
crystal  are  on  the  surface  of  the 
/r  cone  will  be  only  singly  refracted, 
and  emerge  as  a  cylinder,  whose 
direction,  the  normal  to  uu',  is 
called  an  optic  axis. !  As,  however, 
the  wave  surface  is  symmetrical  to 
the  two  axes  in  the  plane  o  x  and 
o  z,  a  line,  o  v,  to  the  left  of  o,  and 
equally  inclined  to  o  z,  will  have  similar  properties,  or  will  also 
be  an  optic  axis,  and  the  crystal  will  be  biaxial  or  biaxal. 

The  section  of  the  crystal  in  which  all  these  axes  lie  is 
called  the  plane  of  the  optic  axes,  and  the  axes  of  elasticity 
which  bisect  the  angles  between  the  latter  are  their  median, 
or  mean  lines,  or  bisectrices.  These  are  further  characterised 
as  first,  or  principal,  and  second,  or  supplementary,  mean 
lines ;  the  former  bisecting  the  acute,  and  the  latter  the 
obtuse  or  supplementary  angle  of  the  optic  axes. 

The  position  of  the  optic  axes,  with  reference  to  the 
median  lines,  depends  upon  the  differences  of  velocity  of 
wave  transmission  in  the  three  axes  of  elasticity.  If  the  mean 
value  is  nearer  to  the  maximum  than  to  the  minimum,  or  the 
radius  o  B  approaches  closer  to  the  major  than  to  the  minor 
semi-axis  of  the  ellipse,  as  in  fig.  334,  the  acute  angle  of  the 

1  In  this  figure  the  divergence  of  the  rays  at  the  apex  of  the  cone  is 
far  greater  than  any  known  example.  The  divergence  is  very  small  in 
all  substances  but  those  with  very  strong  double  refraction,  such  as 
Aragonite,  whose  maximum  and  minimum  indices  differ  by  about  10 
per  cent.,  and  in  which  the  angle  u  o  u'  is  i°  55'. 


CHAP.  XII.] 


Biaxial  Crystals. 


243 


optic  axes  2  v  will  have  the  axis  of  minimum  elasticity  for  their 
first  median  line  ;  but  if  it  is  nearer  to  the  minimum,  or  if  o  B 
is  but  very  little  greater  than  the  semi-axis  oc  (fig.  335), 
the  axis  of  maximum  elasticity  is  the  first  median  line. 
Crystals  of  the  first  class  are  called  positive,  and  those  of  the 
second  negative.  The  value  of  the  angle  -2  v  may  therefore 
vary  within  wide  limits,  but  must  be  always  greater  than  o° 
or  less  than  90°.  It  is  tolerably  constant,  and  often  a  most 


FIG.  334. 


FIG.  335. 


characteristic  element  in  such  minerals  as  are  of  com- 
paratively simple  constitution ;  but  in  those  containing 
numerous  elements,  and  giving  complex  molecular  formulae, 
such  as  mica  felspar  and  topaz,  it  may  vary  very  considerably 
in  different  specimens  of  the  same  mineral,  or  even  in  differ- 
ent parts  of  the  same  crystal. 

The  curves  formed  by  the  intersection  of  the  wave 
surfaces  with  the  other  two  principal  optic  sections  will  be, 
in  the  plane  containing  ox  and  OY  (fig.  336),  an  ellipse 
whose  semi-axes  are  proportional  to  the  maximum  and 
mean  velocities  enclosing  a  circle  whose  radius  is  propor- 
tional to  the  minimum  velocity ;  and,  in  that  containing  o  Y 
and  oz  (fig.  337),  a  circle  whose  radius  is  proportional  to 
the  maximum  velocity  enclosing  an  ellipse  having  its  semi- 
diameters  proportional  to  the  minimum  and  mean  velo- 


244 


Systematic  Mineralogy.          [CHAP.  xii. 


cities.  In  each  of  the  three  sections,  therefore,  one  ray 
will  follow  the  law  of  ordinaiy  refraction,  and  if  the  indices 
corresponding  to  these  be  determined  by  prisms  whose 
refracting  edges  are  respectively  parallel  to  the  three  axes 
of  elasticity,  three  values  will  be  obtained,  which  are  the 
three  principal  refractive  indices.  These  are  usually  distin- 
guished by  the  symbols  a,  /3,  and  y ;  the  first  referring  to 
the  minimum,  the  second  to  the  mean,  and  the  last  to  the 


FIG.  336. 


maximum  value.  For  their  determination  three  prisms  are 
not  absolutely  necessary,  as  two  will  suffice  if  they  are  so 
cut  that  the  refracting  angle  is  either  bisected  symmetrically 
by,  or  one  face  is  parallel  to,  a  principal  optic  section. 
The  three  refractive  indices  and  the  corresponding  axes  of 
elasticity  are  the  optical  constants  of  a  biaxial  crystal  from 
which  the  actual  angles  of  the  optic  axes  are  calculated. 

The  method  described  in  page  234  may  also  be  used, 
but  the  phenomena  are  more  complicated  than  with  isotropic 
media,  the  distance  through  which  the  objective  must  be 
shifted  being  different,  for  ordinary  and  extraordinary  rays, 
from  the  production  of  what  are  called  by  Sorby  '  bifocal ' 
images.  For  the  details  of  the  special  appliances  required 
the  reader  is  referred  to  the  original  memoirs. 


CHAP.  XII.]  Optical  Classification.  245 

The  statements  contained  in  the  preceding  pages  of  this 
chapter  may  be  conveniently  summarised  as  follows  : — 

Supposing  light  to  be  a  succession  of  spherical  waves, 
the  wave  generated  at  each  point  being  spherical,  it  is  the  line 
of  greatest  intensity  which  represents  the  visual  ray,  and  this 
is  ordinarily,  in  a  homogeneous  and  isotropic  medium,  a 
straight  line.  But  when,  in  consequence  of  definite  arrange- 
ment, there  is  not  isotropy,  all  the  waves  are  not  spherical, 
and  the  path  of  the  visual  ray  is  not  a  mere  straight  line, 
speaking  generally.  The  chief  cases  are  : — 

i.  Isotropy  in  general. 

a.  Absolute  amorphism,  such  as  a  homogeneous  glassy 

or  colloid  medium. 

b.  Cubic  symmetry. 

Any  three  pairs  of  equal  forces  at  right  angles  are  in 
equilibrium,  as  in  fig.  338,  the  third  being  perpendi- 
cular to  the  page,  and  may  be  replaced  by  three  other 

FIG.  338.  FIG.  339. 


pairs  of  equal  orthogonal  forces  (fig.  339),  arbitrarily 
chosen  as  regards  aspect  (or  direction  in  space):  any 
equilibrium  which  can  be  represented  by  one  such 
system  is  therefore  isotropic  as  regards  all  forces  of 
the  same  nature. 

Isotropy  about  an  axis. 

a.  Amorphism  polarised  in  one  direction,  as  in  an 
amorphous  elastic  and  isotropic  mass,  subjected  to 
definite  pressure  or  tension. 

/'.  Quadratic  symmetry  in  planes  at  right  angles  to  the 
axis.  The  same  remark  applies  about  two  pairs  of 


246  Systematic  Mineralogy.          [CHAP.  xn. 

orthogonal  forces  in  a  plane,  as  about  three  pairs  in 
solid  in  the  case  of  cubico- symmetry. 

c.  Ternary  plane  symmetry,  or  senary  (fig.  340).  The 
plane  forces  may  be  replaced  by  two  pairs  of  equal 
orthogonal  forces,  as  far  as  mere  equilibrium  is  con- 
cerned. This  symmetry  about  an  axis  constitutes 
the  isotropy  of  uniaxial  systems,  the  isotropic  axis 
being  the  optic  axis. ' 

3.  Symmetry.  If  any  three  pairs  of  forces,  equal  in 
pairs,  keep  a  material  point  in  equilibrium  (fig.  341),  the 
third  pair  lying  out  of  the  page,  they  may  be  replaced  by 

FIG.  340.  FIG.  341. 


three  pairs  of  orthogonal  forces  of  definite  magnitude,  whose 
position  and  magnitude  depend  upon  the  position  and  mag- 
nitude of  the  original  forces.  These  orthogonal  forces  are 
called  the  axes  of  optical  elasticity.  Taking  them  as  the 
axes  of  an  ellipsoid,  the  optic  axes  are  lines  at  right  angles 
to  its  circular  sections. 

The  optical  characters  of  minerals  are  most  strikingly 
evidenced  by  the  interference  of  phenomena  observed  when 
parallel  sections  cut  in  known  directions  are  examined  under 
polarised  light.  The  observation  depends  upon  the  general 
principle  that  the  two  rays  resulting  from  the  passage  of  a 
plane  polarised  ray  through  a  double  refracting  crystal  are 

1  There  is  symmetry  about  a  plane  normal  to  the  axis  in  all  such 
crystals.  It  is  instructive  to  compare  this  with  the  unsymmetrical,  but 
isotropic,  character  of  a  vertical  line  in  the  earth's  atmosphere,  taken 
as  a  refracting  medium.  There  is,  however,  no  polarisation,  as  the 
air  is  not  under  tension  in  any  one  direction  mere  than  another. 


CHAF.  XII.]  Polariscopes.  247 

capable  of  interference  like  ordinary  light  when  reduced  to 
the  same  plane  of  polarisation  by  passing  through  a  second 
polarising  medium.  The  instruments  used  for  this  purpose, 
known  as  polarising  microscopes  or  polariscopes,  differ  some- 
what in  construction,  and  chiefly  as  regards  the  polarising 
agent  employed.  Those  most  commonly  used  are  a  bundle  of 
parallel  glass  plates,  and  Nicol's  prism  ;  the  latter  is  a  cleav- 
age rhombohedron  of  calcite,  whose  length  is  about  twice  its 
breadth,  the  smaller  end  faces  being  ground  down  until  the 
angle  with  the  longer  edges  of  the  other  four  faces  is  reduced 
from  71°  to  68°,  when  the  principal  section  will 
resemble  A  BCD  (fig.  342).  The  crystal  is  then 
sawn  in  half,  on  a  plane  at  right  angles  to  the 
principal  section,  whose  projection  is  the  line  B  c, 
and  the  surfaces,  when  perfectly  polished,  are  ce- 
mented together  parallel  to  the  original  position 
by  a  layer  of  Canada  balsam  ;  the  side  faces  are 
blackened,  and  enclosed  in  a  metal  tube,  the 
ends  being  exposed.  If  a  ray  of  light  meets  one 
of  the  end  faces  in  a  direction  parallel  to  the 
length  of  the  prism,  it  will  be  doubly  refracted  ; 
the  extraordinary  ray,  having  the  lower  refractive 
index — nearly  the  same  as  that  of  Canada 
balsam,  namely  i'536 — passes  through  almost  without  de- 
viation in  the  direction  re  ;  the  ordinary  ray,  on  the  other 
hand,  having  the  higher  refractive  index — 1*654 — is  deflected 
through  a  larger  angle,  and  consequently  meets  the  optically 
lighter  layer  of  balsam  at  such  an  inclination  that  it  is 
totally  reflected,  and  passes  into  the  blackened  covering  at 
o',  where  it  is  absorbed.  Of  the  two  rays,  therefore,  produced 
by  double  refraction,  only  the  extraordinary  ray,  whose 
planes  of  vibration  and  polarisation  are  respectively  parallel 
and  perpendicular  to  the  principal  section,  is  transmitted ; 
and  such  a  ray  will  pass  freely  through  a  second  prism  of 
the  same  kind,  if  the  principal  sections,  represented  by  the 
longer  diagonals  of  the  end  faces,  of  both  are  parallel ;  but 


248  Systematic  Mineralogy.         [CUAP.  XII. 

if  their  principal  sections  are  crossed  at  right  angles,  the 
light  is  completely  extinguished.  For  any  intermediate  azi- 
muth of  vibration,  the  light  transmitted  will  be  the  com- 
ponent of  the  resolved  ray  that  vibrates  parallel  to  the  prin- 
cipal section,  and,  therefore,  will  be  less  in  proportion  as  the 
angle  between  the  principal  section  of  the  Nicol  and  the 
plane  of  vibration  of  the  ray  increases  ;  and  the  same 
relation  holds  good  for  two  plane  polarised  rays,  such  as 
originate  by  the  passage  of  ordinary  light  through  a  doubly- 
refracting  medium,  vibrating  in  different  planes,  and  entering 
Jhe  prism  simultaneously.  In  the  latter  case,  the  portions 
of  the  rays  transmitted  are  said  to  be  reduced  to  the  same 
plane  of  polarisation,  and  are  in  condition  to  interfere  in  the 
same  manner  as  rays  of  ordinary  unpolarised  light. 

Fig.  343  is  a  generalised  representation  of  the  common 
form  of  polariscope.  It  consists  of  two  Nicol  prisms,  A  and 
B,  attached  to  a  vertical  pillar,  with  a  plane  mirror  for  re- 
flecting light  in  the  lower  prism,  and  a  carrier  D,  for  the 
crystal  under  examination,  which  may  be,  either  fixed  to  the 
pillar,  or,  as  is  more  generally  the  case,  mounted  in  a  divided 
ring  which  rotates  on  the  top  of  the  case  of  the  lower  prism. 
The  latter,  which  is  called  the  polariser,  is  fixed,  and  is 
usually  of  a  larger  size  than  the  upper  one,  or  analyser, 
which  is  movable,  both  about  its  own  axis  and  in  a  vertical 
direction,  by  a  rack-work  fixed  to  the  pillar.  This  arrange- 
ment is  used  for  the  examination  of  minerals  under  parallel 
light,  the  whole  of  the  rays  passing  through  the  instrument 
being  parallel  to  its  axis  and  to  each  other  ;  but  to  produce 
the  characteristic  phenomena  which  are  observed  when  the 
crystal  is  traversed  by  rays  of  different  degrees  of  obliquity, 
a  combination  of  condensing  lenses  is  placed  above  the 
polarising  prism,  and  a  corresponding  combination  in  the 
reverse  direction  below  the  analyser,  which  are  so  adjusted 
that  their  common  focal  point  is  in  the  centre  of  the  plate 
of  mineral  under  investigation.  These  are  represented  in 
section  in  fig.  344  ;  the  lower  series  of  four  plano-convex 


CHAP.  XII.] 


Polariscopes. 


249 


lenses,  c^  c.2,  cz,  c±,  form  the  condenser,  and  the  upper,  0,,  o.2 
03,  04,  the  objective.  These  combinations  are  equivalent  to 
single  lenses  of  short  focal  length  and  low  magnifying 

FIG.  344. 


power,  but  giving  a  large  field  of  view.  The  size  of  the 
image  may  also  be  amplified  by  a  double  convex  lens, 
forming  an  eye-piece,  placed  below  the  analysing  prism. 

When  a  parallel-sided  plate  of  a  transparent  isotropic 
substance  is  placed  upon  the  carrier  D,  it  will  have  no  effect 
upon  the  light  passing  through  the  instrument — that  is,  if  the 
planes  of  vibration  of  the  two  Nicols  be  parallel,  the  field 
will  be  illuminated ;  if  they  be  crossed  at  right  angles,  there 
will  be  complete,  and  at  any  other  angle  partial  obscurity, 
exactly  the  same  as  there  would  be  by  the  mere  action  of 
the  prisms  themselves  ;  and  further,  no  change  in  the  light 
will  be  observed  when  the  plate  is  turned  round  in  its  own 
plane.  These  conditions,  which  apply  both  to  amorphous 
substances  and  those  crystallising  in  the  cubic  system,  hold 
good  both  for  parallel  and  convergent  light — such  substances 
being  said  to  be  without  depolarising  effect. 


250  Systematic  Mineralogy.  [CHAP.  XII. 

If  a  plate  of  a  uniaxial  crystal,  which  may  be  either  an 
artificial  section  cut  perpendicularly  to  the  optic  axis,  a  suf- 
ficiently thin   crystal  with  well-developed  basal   pinakoid 
faces,  or  a  basal  cleavage  plate,  be  examined  in  parallel 
polarised  monochromatic  light,  it  will  appear  light  or  dark 
according  to  the  position  of  the  Nicol's 1  prisms,  in  the  same 
way  as  an  isotropic  substance,  the  direction  followed  by  the 
light  being  that  of  single  refraction  in  the  crystal.     If,  how- 
ever, the  section  be  cut  parallel  to  the  optic  axis,  it  will 
when  placed  between  parallel  Nicols  appear  bright  when  its 
principal  section  corresponds  to  their  planes  of  vibration, 
and  if  turned  in  its  own  plane  the  light  will  diminish  with 
the  angular   deviation  and  be  completely  extinguished  at 
45°,  from  which  point  the  light  will  again  increase  to  a 
maximum  at  90°,  disappear  at  135°,  and  so  on  through  the 
complete  rotation,  the  field  of  the  instrument  being  four 
times  light  and  dark  alternately.      When  the   Nicols  are 
crossed,  the  light  is  extinguished,  when  the  principal  section 
of  the  plate  is  parallel  to  the  plane  of  vibration  of  either 
Nicol,  or  at  the  azimuths  o°,  90°,  180°,  and  270°,  and  is 
brightest  at  the  intermediate  or  half- quarter  points  of  the 
circle,  45°,  135°,  &c.     There  are,  therefore,  four  alternations 
of  light  and  darkness  during  the  revolutions  of  the  plate,  as 
before,  but  the  order  of  their  occurrence  is  inverted.    When 
white  light  is  used  and  the  plate  is  sufficiently  thin  to  cause 
a  retardation  of  one  of  the  refracted  rays  corresponding  to  a 
half-wave  length  of  any  particular  colour,  that  colour  will  be 
extinguished,  and  consequently,  at  the  position  of  maximum 
illumination,  the  plate  will  show  the  residual  or  comple- 
mentary tint ;  or  if  a  spectroscope  be  used  over  the  analyser, 
the  spectrum  will  show  an  absorption  band  corresponding 
to  the  missing  colour.     These  positions  with  crossed  Nicols 
are  at  the  half-quarter  points,  but  with  a  plate  whose  retar- 
dation is  a  complete  wave  length,  these  points  correspond  to 

1  Or,  as  they  are  usually  called,  '  Nicols.' 


CHAP.  XII.]  Uniaxial  Polarisation.  251 

those  of  extinction,  the  greatest  illumination  being  at  o°, 
90°,  1 80°,  and  270°.  When  the  plate  is  much  thicker, 
so  that  the  retardation  amounts  to  f,  f,  £,  or  a  greater 
number  of  half-wave  lengths,  so  much  colour  is  extinguished 
by  interference  that  the  residual  tint  is  scarcely  distinguish- 
able from  white,  forming  the  so-called  white  of  the  higher 
order,  and  therefore  the  plate  shows  four  times  light  and 
dark  in  a  revolution,  nearly  in  the  same  way  as  in  homo- 
geneous light.  In  these  cases  the  spectrum  of  the  plate 
will  be  crossed  by  several  dark  bands  ;  these  are  especially 
well  seen  in  Quartz.  When  the  Nicols  are  parallel,  the 
colours  seen  at  the  points  of  maximum  illumination  are 
complementary  to  those  obtained  when  they  are  crossed. 

Under  convergent  homogeneous  light,  the  appearances 
presented  by  a  section  perpendicular  to  the  optic  axis 
between  parallel  Nicols  are  seen  FIG.  345. 

in  fig.  345.  A  central  bright  space 
corresponding  to  the  direction  of 
the  optic  axis,  or  that  of  single 
refraction,  is  surrounded  by  seg- 
ments  of  concentric  dark  rings, 
which  are  broadest  along  lines 
45°  and  135°  from  the  plane  of 
vibration  of  the  Nicols,  and  fade 
gradually  towards  the  lines  of  o°  and  90°,  producing  the 
effect  of  a  bright  four-armed  cross,  traversing  the  rings  sym- 
metrically. The  dark  rings  nearest  to  the  centre  correspond 
to  the  positions  in  which  the  rays  produced  by  double  re- 
fraction in  the  crystals  emerge,  with  a  difference  of  half  a 
wave  length,  and  therefore,  when  reduced  to  the  same  plane 
of  polarisation  by  the  analyser,  interfere  and  produce  com- 
plete extinction  on  the  line  of  the  principal  section,  dd'. 
In  any  other  principal  section  making  a  smaller  or  larger 
angle  with  the  line  E  w,  light  incident  at  the  Same  angle  will 
be  similarly  refracted  and  retarded,  but  the  rays  will  not  be 
equally  resolved  by  the  analyser  ;  that  one  whose  direction 


252  Systematic  Mineralogy.  [CHAP.  XII. 

of  vibration  is  nearest  to  E  w  being  more  completely  trans- 
mitted than  the  other,  which  vibrates  at  right  angles  to  it ; 
and  therefore,  although  there  will  still  be  interference,  the 
extinction  will  not  be  complete,  owing  to  the  unequal  ampli- 
tude of  the  two  rays  transmitted  by  the  analyser.  This 
difference  is  at  a  maximum  in  the  principal  sections  that 
are  respectively  parallel  and  perpendicular  to  EW,  as  in 
these  only  one  of  the  refracted  rays  is  transmitted  by  the 
analyser,  and  therefore  the  field  remains  bright,  forming  the 
arms  of  the  cross.  With  increased  distance  from  the  centre, 
as  the  obliquity  of  the  incident  light  increases,  the  paths  of 
the  refracted  rays  in  the  crystal  are  lengthened,  and  their 
difference  of  phase  at  emergence  becomes  greater,  so  that  at 
a  point  some  distance  beyond  the  first  dark  ring,  where  it 
amounts  to  two  half-wave  lengths,  the  light  is  augmented 
by  their  interference  producing  a  bright  ring.  Beyond  this 
it  diminishes  again,  total  obscurity  being  produced  when 
the  difference  of  phase  is  f ,  and  so  on,  the  field  being  bright 
and  dark  from  the  interference  of  waves  differing  in  phase 
by  even  and  uneven  numbers  of  half- wave  lengths  respec- 
FIG.  346.  tively.  With  increased  obliquity  of 

the  light,  however,  these  interferences 
recur  more  rapidly,  and  therefore 
towards  the  border  of  the  field  of  view, 
the  rays  are  narrower  and  closer 
together  than  in  the  centre.  If  the 
plate  be  viewed  between  crossed 
Nicols,  the  complementary  phenomena 
shown  in  fig.  346  are  observed  :  a 
series  of  continuous  dark  rings  taking  the  places  of  the 
bright  ones  in  fig.  345,  with  a  dark  rectangular  cross,  one 
bar  of  which  corresponds  to  the  projection  of  the  plane  of 
vibration  of  the  polariser,  and  the  other  to  that  of  the 
analyser,  crossing  them  symmetrically. 

In  white  light,  the  rings,  instead   of  being   dark,  are 
coloured,  the  tint  of  any  one  corresponding  to  the  residual 


CHAP.  XII.]  Umaxial  Polarisation.  253 

colour  after  extinction  of  the  particular  light  which  is  re- 
tarded by  a  half-wave  length  in  one  of  the  refracted  rays. 
As  these  are  constant  for  the  same  angle  of  incidence,  the 
colour  will  be  the  same  at  any  similar  distance  from  the 
centre  of  the  plate,  from  which  circumstance  the  rays  are 
called  isochromatic  curves.  In  uniaxial  crystals  these  are 
circles  if  the  section  be  cut  perfectly,  or  very  nearly, 
square  to  the  optic  axis  ;  but  if  it  is  sensibly  oblique,  they 
will  be  somewhat  elliptical,  and  the  cross,  instead  of  changing 
from  black  to  white  by  the  rotation  of  the  analyser,  will  be 
resolved  into  a  looped  figure  like  that  of  a  biaxial  crystal. 

The  colours  of  the  rings  recur  in  the  order  of  the  New- 
tonian interference  scale,  those  of  higher  refrangibility  are 
generally  seen  only  in  the  rings  nearest  the  centre,  alter- 
nations of  red  and  green  of  diminishing  intensity  appearing 
towards  the  margin  of  the  field.  These  latter,  the  so-called 
colours  of  higher  orders,  can  only  be  seen  within  much 
narrower  limits  than  the  black  rings  produced  in  monochro- 
matic light,  and  therefore  the  figures  produced  with  the 
latter  are  much  sharper  and  better  defined  than  with  white 
light  The  distance  between  the  rings  varies  with  the  strength 
of  the  double  refraction,  or  the  difference  between  the  refrac- 
tive indices,  of  the  ordinary  or  extraordinary  rays,  the  thick- 
ness of  the  plate,  and  the  refrangibility  of  the  light  used. 
For  plates  of  equal  thickness,  of  different  substances,  they 
will  appear  closest  in  those  crystals  that  have  the  stronger 
double  refraction  and  higher  refractive  power.  For  the 
same  substance  the  distance  between  the  rings  will  be  in- 
creased by  diminishing  the  thickness  of  the  plate,  and  to 
a  less  degree  by  illuminating  with  less  refrangible  light. 
Figures  345,  346,  represent  the  rings  seen  in  a  section  of 
Calcite  ^  inch  thick,  in  nearly  monochromatic  red  light,  pro- 
duced by  passing  solar  light  through  a  plate  of  ruby  glass  ; 
but  if  a  blue  light  be  used  in  the  same  way,  the  rings  will  be 
crowded  much  closer  together,  but  will  become  invisible  at 
about  two-thirds  of  the  former  breadth  of  field.  When  the 


254  Systematic  Mineralog}'.          [CHAP.  xil. 

section  is  very  thin,  Jjjth  of  an  inch  or  less,  no  rings,  but 
only  a  black  cross  or  four  black  spots  are  seen,  according 
to  the  position  of  the  analyser.  With  minerals  of  weak 
double  refraction,  such  as  Idocrase  or  Apophyllite,  thicker 
sections  are  required  in  order  to  see  the  rings. 

The  method  of  determining  the  sign  of  the  double 
refraction  uniaxial  crystals  will  be  described  subsequently. 

Circular  Polarisation.  A  section  of  a  quartz  crystal  per- 
pendicular to  the  optic  axis  viewed  by  monochromatic  light 
between  crossed  Nicols,  unlike  other  uniaxial  crystals,  allows 
some  light  to  pass  through,  and  it  is  only  by  turning  the 
analyser  through  a  certain  angle  that  the  light  can  be  extin- 
guished ;  but  when  this  position  has  been  determined  the 
plate  will  remain  dark  when  turned  in  its  own  plane  in  the 
same  manner  as  an  ordinary  uniaxial  crystal,  showing  that 
the  light  enters  the  analyser  in  the  condition  of  plane  polar- 
isation, but  that  the  direction  of  its  plane  of  vibration  has 
been  changed.  This  condition  of  the  light,  called  circular 
polarisation,  is  produced  by  all  substances,  whether  isotropic 
or  uniaxial,  that  crystallise  in  plagihedral  forms — that  is,  the 
tetartohedral  forms  of  the  cubic,  the  trapezohedral  tetarto- 
hedra  of  the  hexagonal,  and  the  plagihedral  hemihedra  of 
the  tetragonal  systems.  Examples  of  the  first  and  last  of 
these  are  only  known  in  artificially  crystallised  salts,  but  the 
property  is  most  strikingly  developed  in  the  hexagonal 
system,  in  the  minerals  Quartz  and  Cinnabar.  The  angular 
deviation  of  the  plane  of  polarisation  produced  by  a  plate  of 
Quartz  increases  with  its  thickness,  as  well  as  with  the  re- 
frangibility  of  the  light  used  for  illuminating,  being  greatest 
for  the  blue  and  least  for  the  red  end  of  the.  spectrum,  or 
for  any  particular  thickness  it  varies  inversely  as  the  square 
of  the  wave  length.  With  a  quartz  plate  of  ^th  of  an  inch 
thick,  the  rotations  are  : — 

Red  ig'oo0,  Yellow  24*00°,  Blue  32°,  extreme  Violet  44°. 
The  specific  rotatory  power  of  Cinnabar  is  very  much 


CHAP.  XII.]  Circular  Polarisation.  255 

greater,  being,  according  to  Descloizeaux,  fifteen  times  that 
of  Quartz.  As  a  consequence  of  the  difference  in  rotation 
for  different  colours,  a  circularly  polarising  crystal  can  never 
appear  dark  in  parallel  polarised  white  light,  whatever  may 
be  the  .relative  positions  of  the  polariser  and  analyser.  When 
the  latter  are  crossed,  the  rays  whose  plane  of  vibration  most 
nearly  coincide  with  that  of  the  analyser  will  be  most  com- 
pletely transmitted,  while  those  making  larger  angles  with  it 
will  for  the  most  part  be  extinguished.  The  prevailing 
colour  of  the  plate,  therefore,  will  be  mainly  that  of  the 
freely  transmitted  rays,  and  this  will  be  constant  for  any 
plate  of  the  same  thickness.  By  turning  the  analyser  its  plane 
of  vibration  becomes  successively  parallel  to  those  of  the 
different  coloured  rays,  and  the  colour  of  the  field  changes 
with  each  coincidence.  The  order  in  which  the  colours 
meet  each  other  is  not  the  same  in  all  cases,  and  to  produce 
them  in  the  order  of  their  refrangibility  from  red  to  violet 
it  is  sometimes  necessary  to  turn  the  analyser  from  right 
to  left,  and  sometimes  the  reverse.  These  differences 
correspond  to  differences  in  the  position  of  the  plagihedral 
faces  in  the  crystals  from  which  the  plates  are  cut,  the 
former  indicating  right-handed  and  the  latter  left-handed 
forms.  In  some  instances,  however,  the  sections  show  more 
or  less  irregular  patches,  some  having  right-  and  others  left- 
handed  rotation,  proving  that  the  crystals  from  which  they 
are  derived  are  complex  twin  structures. 

In  convergent  polarised  white  light,  Quartz  gives  a  series 
of  isochromatic  rings,  barred  by  black  or  bright  arms,  ac- 
cording to  the  position  of  the  analyser,  in  the  outer  part  of 
the  field ;  but  in  the  centre  where  the  light  is  either  parallel 
or  but 'slightly  inclined  to  the  optic  axis,  the  conditions  are 
similar  to  those  described  for  parallel  light,  and  therefore, 
instead  of  a  complete  cross,  the  central  space  within  the  first 
dark  ring  will  have  the  colour  due  to  the  thickness  of  the 
plate,  when  the  Nicols  are  crossed,  and  will  change  when  the 
analyser  is  turned  as  in  parallel  light.  With  a  right-handed 


256  Systematic  Mineralogy.          [CHAI-.  XII. 

plate  the  rotation  of  the  analyser  to  the  right  causes  an 
apparent  expansion  of  the  rings,  and  the  change  from  a  light 
to  a  dark  field,  and  to  the  left  a  corresponding  contraction  ; 
with  a  left-handed  plate  the  reverse  conditions  prevail. 

When  two  plates  of  Quartz,  perpendicular  to  the  optic 
axis  of  the  same  thickness,  but  of  opposite  rotation,  are 
superposed  in  paralled  polarised  light,  the  rotatory  power  of 
one  will  neutralise  that  of  the  other,  or  the  field  will  remain 
dark  as  with  an  ordinary  uniaxial  crystal.  When  the  two 
sections  are  of  unequal  thickness,  such  as  the  parallel-sided 
plate  obtained  by  the  supeqoosition  of  two  similar  wedges 
inclined  in  opposite  directions,  there  will  be  a  dark  line  or 
neutral  band  where  the  thicknesses  are  exactly  alike,  on 
either  side  of  which  the  characteristic  colour  due  to  the 
preponderance  of  the  thickest  plate  at  that  point  will  appear. 
In  convergent  light,  the  interference  figures  known  as  Airy's 
FIG.  347.  spirals  are  produced,  which,  in  addition 

to  the  concentric  rings  of  an  ordinary 
uniaxial  crystal,  show  a  series  of  four 
spiral  arms  crossing  in  the  centre  of 
the  field,  the  direction  in  which  the 
arms  are  coiled  being  determined  by 
that  of  the  rotation  of  the  lower  plate. 
Thus,  in  fig.  347,  the  left-handed  plate 
is  below  the  right-handed  one ;  but  if  it 
were  above,  the  arms  would  coil  in  the  opposite  direction, 
the  figure  being  otherwise  unaltered. 

These  phenomena  are  sometimes  seen  in  plates  cut  from 
apparently  simple  Quartz  crystals,  proving  them  to  be  parallel 
alternations  of  right-  and  left-handed  ones. 

The  property  of  circular  polarisation  in  Quartz  crystals 
is  confined  to  the  direction  of  the  optic  axis ;  in  every  other 
direction,  therefore,  they  behave  like  ordinary  plane  polaris- 
ing crystals.  Even  in  sections  normal  to  the  axis  a  certain 
thickness  is  required  to  produce  a  sensible  rotation,  a  very 
thin  one  shows  merely  a  black  or  dark  grey  cross,  between 
crossed,  and  four  dark  blue  spots  with  parallel,  Nicols. 


CHAP.  XII.]          Biaxial  Polarisation.  257 

In  the  few  known  cases  of  cubic  tetartohedral  crystals 
the  circular  polarisation  is  independent  of  direction,  and 
any  section  of  the  crystal  shows  the  rotation  of  the  plane  of 
polarisation  equally  well.  The  principal  salts  in  which  this 
has  been  observed  are  the  Bromate  of  Sodium,  Chlorate  of 
Sodium  and  Nitrate  of  Barium.  The  last  two  have  been 
shown  to  be  geometrically  tetartohedral,  but  in  the  first  the 
development  is  inferred  from  the  circular  polarisation. 

Optical  examination  of  biaxial  crystals.  In  parallel 
polarised  light,  a  section  of  a  biaxial  crystal,  when  rotated 
between  crossed  Nicols,  will  appear  four  times  dark  and  light 
(or  coloured  if  sufficiently  thin)  in  each  revolution,  except  it 
be  perpendicular  to  an  optic  axis,  when  it  will  give  a  dark 
field  in  all  positions.  This  latter  case,  however,  rarely 
arises  in  practice,  as  such  sections  are  not  easily  prepared. 
In  the  rhombic  system  the  crystallographic  axes  coincide  _ 
with  those  of  optic_elasticijy,  and  therefore  when  a  section  of  ' 
a  rhombic  crystal  parallel  to  either  a  pinakoid  or  a  prismatic 
face,  whether  prism  or  dome,  is  placed  between  crossed 
Nicols,  the  field  will  appear  dark  whenever  a  crystallo- 
graphic axis  in  the  section  is  parallel  to  the  plane  of  vibra- 
tion of  either  Nicol,  as  well  in  white  as  in  homogeneous  light 
of  any  colour.  In  such  cases  the  directions  of  extinction  are 
said  to  be  parallel  to  the  crystallographic  axes. 

In  the  oblique  system  only  one  axis  of  optic  elasticity  coin- 
cides with  an  axis  of  form — namely,  with  the  orthodiagonal, 
the  other  two  lying  in  the  plane  of  symmetry  at  right  angles 
to  each  other,  but  oblique  to  the  crystallographic  axes  in 
that  plane.  If  therefore  a  thin  tabular  crystal  or  a  section 
parallel  to  either  the  base  or  the  orthopinakoid  be  placed 
between  crossed  Nicols,  it  will  extinguish  the  light  in  the 
same  manner  as  a  rhombic  one — i.e.,  parallel  to  the  crystallo- 
graphic axes  ;  but  if  the  section  be  parallel  to  the  clinopina- 
koid,  it  will  only  appear  dark  when  the  vertical  axis  of 
the  crystal  is  inclined  to  the  vibration  plane  of  a  Nicol  at 
some  angle  which  will  differ  with  the  colour  of  the  light 

s 


258  Systematic  Mineralogy.          [CHAP.  XII. 

employed,  a  property  known  as  the  dispersion  of  the  axes 
of  elasticity.  If  the  section  be  parallel  to  a  face  of  a  prism, 
the  same  kind  of  dispersion  will  be  observed,  but  in  a  dif- 
ferent degree,  the  angle  between  the  vertical  axis  and  the 
direction  of  extinction  being  less  for  the  same  colour  than 
in  the  plane  of  symmetry,  and  these  directions  approach 
nearer  to  parallelism,  as  the  angle  of  the  plane  of  the  sec- 
tion with  the  orthopinakoid  diminishes — that  is,  as  the  front 
or  obtuse  angle  of  the  prism  increases.  In  the  triclinic 
system,  there  being  no  coincidence  in  direction  between  the 
axes  of  form  and  optic  elasticity,  the  directions  of  extinction 
in  any  plane  will  be  oblique  to  the  axes  of  the  crystal. 

The  directions  of  extinction,  therefore,  afford  a  means 
of  determining  the  character  of  the  symmetry  of  a  crystal, 
or  its  crystallographic  system,  and  as  such  their  observa- 
tion is  one  of  the  most  important  operations  in  physical 
crystallography.  An  exact  determination,  however,  is  a 
matter  of  some  nicety,  and  cannot  practically  be  made 
directly,  as  the  light  fades  so  gradually  near  the  point 
of  maximum  darkness  that  the  difference  corresponding  to 
a  change  of  azimuth  of  2  to  3  degrees  is  not  easily  ap- 
preciable. An  indirect  class  of  observation  is  therefore 
adopted,  in  which,  instead  of  the  extinction  of  light,  changes 
in  the  appearance  of  the  interference  cross  of  a  plate  of 
Calcite  normal  to  the  optic  axis  when  viewed  between 
crossed  Nicols,  with  a  plate  of  the  mineral  under  examination 
interposed,  are  used  as  a  test  of  the  change  of  position  of  a 
known  crystallographic  line  in  the  latter.  The  instrument 
for  this  purpose,  called,  from  the  nature  of  the  observation, 
a  stauroscope,1  in  the  original  form  given  to  it  by  Dr.  Von 
Kobell  in  1855,  is  essentially  a  polariscope,  arranged  for 
parallel  light  with  a  Calcite  plate  fixed  below  the  analyser, 
the  latter  being  crossed  to  the  polariser  so  as  to  give  a  perfect 
dark  cross.  Above  the  polariser  is  a  perforated  metal  plate, 

1  Figured  and  described  at  length  in  Rutley's  treatise  on  rocks  in 
this  series. 


CHAP.  XII.]  Stauroscope.  259 

forming  an  object  carrier,  connected  with  a  rotating  divided 
ring  ;  lines  are  ruled  on  the  plate  parallel  to  the  vibration 
plane  of  the  polariser,  and  the  section  of  the  crystal  is  ce- 
mented to  it  with  a  known  edge,  preferably  one  parallel  to 
the  vertical  axis,  as  nearly  parallel  to  one  of  the  lines  as 
possible.  It  is  then  placed  on  the  instrument,  and  adjusted 
to  the  zero  point  of  the  divided  ring,  when,  if  the  edge  is 
parallel  to  an  axis  of  elasticity,  there  will  be  no  change  in 
the  form  of  the  cross ;  but  if  these  lines  are  appreciably  in- 
clined to  each  other,  the  figure  will  appear  to  be  distorted, 
and  it  will  be  necessary  to  turn  the  divided  plate  through  a 
certain  angle  in  order  to  bring  it  back  to  its  proper  form. 

The  results  obtained  by  this  may  be  regarded  as  accurate 
to  within  one  degree  or  thereabouts.  A  more  sensitive 
test  has  been  proposed  by  Brezina,  who  uses  a  compound 
plate  formed  of  two  sections  of  Calcite  whose  planes  are 
not  perfectly  at  right  angles  to  the  principal  axis,  so  joined 
that  the  optic  axes  of  both  lie  in  the  same  plane,  but  crossing 
each  other  in  direction.  This  gives  an  interference  figure 
made  up  of  peculiar  curves,  the  essential  element  being  an 
elliptical  ring  lying  with  its  longer  axis  right  and  left,  and 
divided  symmetrically  by  a  vertical  bar  which  extends  to  the 
limit  of  the  field  of  view  in  a  continuous  line,  when  the  Nicols 
are  exactly  crossed;  but  avery  slight  disturbance  produced  by 
a  crystal  whose  planes  of  vibration  are  not  parallel  to  those 
of  the  instrument  produces  a  break  in  the  line,  the  part 
within  the  ring  becoming  inclined  to  those  above  and  below 
it,  the  change  being  said  to  be  appreciable  with  an  angular 
deviation  of  a  very  few  minutes. 

In  all  cases  stauroscopic  observations  must  be  made  with 
monochromatic  light,  the  dispersion  of  the  axes  varying  with 
light  of  different  colours. 

In  convergent  polarised  light  a  section  of  a  biaxial 
crystal  perpendicular  to  an  optic  axis  between  crossed  Nicols 
shows  a  series  of  nearly  circular  rings  whose  intervals,  as 
in  the  cases  already  considered,  depend  upon  the  thickness 


260  Systematic  Mineralogy.  [CHAP.  XII. 

of  the  plate  and  the  strength  of  the  double  refraction,  crossed 
by  a  single  dark  line  instead  of  the  four-armed  cross  seen 
in  uniaxial  crystals,  and  which  revolves  with  the  rings  when 
the  plate  is  turned  on  its  own  plane.  When  the  section  is 
taken  perpendicular  to  the  first  median  line,  the  Nicols  being 
crossed  and  the  line  corresponding 
to  the  projection  of  the  optic  axial 
plane  parallel  to  the  vibration  plane 
of  one  of  them,  the  system  of  rings, 
represented  in  fig.  348,  will  be  seen. 
These  are  curves  known  as  lemnis- 
cates,  the  points  where  the  optic  axes 
meet  the  surface  of  the  plate  or  their 
poles  are  the  centres  of  indepen- 
dent series  of  rings,  the  innermost, 

or  polar  rings,  being  nearly  circular,  while  the  succeeding 
ones  are  lengthened  progressively  towards  the  opposite 
axis,  until  about  third  or  fourth  they  meet,  forming  the 
cross-looped  or  figure  of  8,  the  crossing  points  of  the  loops 
making  the  pole  of  the  median  line.  Beyond  this  they  are 
continuous  round  both  axes,  with  a  slight  compression  in 
the  centre,  which  is  less  marked  in  each  succeeding  ring,  so 
that  those  at  the  outside  of  the  visible  field  are  very  similar 
in  appearance  to  ellipses.  The  lines  corresponding-  to  the 
vibrajjnn  plages  of  the  Nicols  are,  as  in  the  case  of  uniaxial 
crystals,  marked  by  a  ^darFjrectangular Derosa,  but  the  bars? 
of  unequal  intensity,  that  joining  the  optic  axes  being  as  a 
rule  darker  and  b^ter  defined  than  the  vertical  one. 

When  the  plate  is  turned  through  45°  in  its  own  plane, 
the  position  of  the  Nicols  being  unchanged,  the  rings  keep 
their  relative  positions,  though  revolving  with  the  plate,  but 
the  cross,  as  seen  in  fig.  349,  is  resolved  in  two  dark  bands, 
usually  sharply  defined  where  they  cross  the  central  rings, 
but  becoming  wider  and  indistinct  towards  the  extremities  of 
the  field.  These  bands,  or,  as  they  are  usually  called, 
*  brushes,'  are  portions  of  the  two  branches  of  an  hyperbola, 


CHAP.  XII.]  Biaxial  Interference  Figures.          261 


FIG.  349. 


and  the  points  where  they  cross  the  line  of  the  optic  axial 
plane,  mark  the  extremities  of  the  optic  axes.  The  dis- 
tance between  the  summits  of  the 
curves,  when  at  their  widest  separa- 
tion, is  an  indication  of  the  amount 
of  the  inclination  of  the  optic  axes, 
and  if  the  field  of  the  instrument  be 
provided  with  a  micrometer  scale  of 
equal  parts,  the  angle  between  them 
may  be  approximately  measured. 

By  reducing  the  thickness  of  the 
plate,  the  distance  traversed  by  the 

light  is  so  much  shortened  that  only  the  rays  of  more 
oblique  incidence  have  sufficient  length  of  path  in  the 
crystal  to  emerge  with  the  difference  of  phase  (^  A)  neces- 
sary to  produce  extinction  of  light  by  interference,  and 


FIG.  350. 


FIG.  331. 


therefore  the  rings  in  such  a  plate  will  be  further  apart,  and 
the  innermost  one  will  be  at  a  greater  distance  from  the 
centre  than  in  a  thicker  plate,  the  polar  rings  and  the 
looped  figure  generally  disappearing,  but  the  cross  and 
brushes  will  not  be  altered.  The  character  of  these  figures 
is  seen  in  figs.  350  and  351,  the  first  corresponding  to  the 
position  in  fig.  348,  and  the  second  to  that  in  fig.  349.  The 
same  kind  of  figure  is  seen  with  thick  plates  when  the  double 
refraction  of  the  mineral  is  weak. 


262 


Systematic  Mineralogy. 


[CHAP.  XII. 


Figs.  352-354  represent   the  successive  disappearance 
of  the  inner  rings  in  films  of  mica,  when  the  thickness  is 


FIG.  352. 


FIG.  353. 


FIG.  354- 


reduced.     Fig.  352  is  the  figure  seen  in  a  plate  producing  a 
retardation  of  phase  of  £  X,  fig.  353  of  f  X,  fig.  354  of  f  X, 


FIG.  355. 


FIG.  356. 


fig.  355  of  £X,  and  fig.  356  of  \\.  In  the  last,  scarcely 
any  portion  of  the  rings  is  seen,  but  the  positions  of  the 
hyperbolic  arms  or  bj^ies  are  equally  well  marked  in  all. 

When  the  angleV>f  the  axes  is  very  small,  the  figure  is 
often  scarcely  distinguishable  from  that  of  a  uniaxial  crystal, 
the  rings  being  nearly  circular,  and  the  four  arms  of  the  cross 
similar  in  intensity.  The  difference,  however,  becomes 
apparent  on  rotating  the  plate,  when  the  arms  divide  into 
hyperbolic  branches,  instead  of  remaining  unchanged,  as 
they  should  do  if  the  substance  were  actually  uniaxial. 

The  exact  measurement  of  the  angle  of  the  optic  axes  is 
effected  by  a  polarising  instrument  with  a  goniometer 
attached  to  it,  a  rod  in  the  prolongation  of  the  axis  of  the 


CHAP.  XII.]     Apparent  and  Real  Axial  Angles.         263 

latter  forms  the  object-carrier,  and  allows  the  plate  to  be 
turned  about  an  axis  in  its  own  plane.  The  instrument  is 
placed  horizontally,  with  the  planes  of  vibration  of  the  prisms 
crossed,  and  the  plane  of  the  optic  axes  so  arranged  as  to 
show  the  maximum  separation  of  the  brushes.  The  plate 
is  then  turned  about  its  axis  of  suspension  by  the  rod  so  as 
to  bring  each  of  the  optic  axes  into  the  centre  of  the  field 
by  intersecting  the  cross  lines  of  the  micrometer  succes- 
sively, the  vernier  arm  of  the  goniometer  being  read  for  each 
position  ;  the  difference  between  these  readings  is  the  ob- 
served angle  between  the  optic  axes,  or  twice  that  between 
either  one  and  the  first  median  line. 

This  angle  so  obtained,  called  the  apparent  angle  of  the 
optic  axis,  and  indicated  by  the  symbol  2  E,  would  be  the 
same  as  the  true  angle  2  V,  found  by  calculation  from  the 
three  principal  refractive  indices,  by  the  formulae  on  p.  276,  if 
the  crystal  plate  were  shaped  either  to  a  sphere  having  the 
first  median  line  for  a  polar  axis,  or  to  a  cylinder  with  its 
axis  parallel  to  that  of  its  mean  elasticity,  in  either  of  which 
cases  the  rays  emerge  perpendicularly  to  the  surface  of  sepa- 
ration, and  therefore  preserve  their  direction  in  passing  from 
the  crystal  to  the  air.     In  all  other  cases  there  will  be 
sensible  deviation  in  the  rays  passing  from  the  denser  to 
the  lighter  medium,  and  therefore  the  apparent,  will  always 
be  greater  than  the  true  angle  of  the  optic  axis,  the  amount 
of  difference  increasing  with  the^^lute  value  of  the  latter, 
and  the  specific  refractive  enen^^^the  crystal.     We  have 
already  seen  that  of  two  rays  ongmating  by  double  refrac- 
tion, whose  paths  are  in  the  optic  axial  plane,  one,  vibrating 
parallel  to  the  axis  of  mean  elasticity,  is  in  the  condition  of 
an  ordinary  ray  having  the  constant  refractive  index  ft  in 
any  direction,  while  the  other  is  an  extraordinary  ray  in  all 
directions  but  those  of  the  optic  axes,  where  its  index  also 
=  /3.     It  is  only  requisite,  therefore,  to  know  the  value  of 
the  mean  refractive  index  of  the  substance  in  order  to  con- 
vert the  apparent  into  the  real  angle  of  the  optic  axes,  or 


264  Systematic  Mineralogy.  [CHAP.  xn. 

the  reverse  ;  the  relation  of  the  angles  Fand  JE,  or  those  of 
an  optic  axis  to  the  median  line,  being  expressed  by 


sin.  E  =  ft  sin.  V,  and  sin.  V=  —jr~- 

As  the  real  angle  of  the  optic  axes  may  have  any  value 
up  to  90°,  it  will  often  be  found  that  the  apparent  angle 
exceeds  a  right  angle,  so  that  it  is  not  always  possible  to 
see  the  rings  or  brushes  about  both  axes  at  once  :  the  field 
of  the  polarising  instrument,  as  ordinarily  constructed,  taking 
in  about  125°  or  135°  at  the  most.  It  also  frequently 
happens  with  crystals  whose  optic  axes  are  very  divergent, 
especially  when  their  refractive  power  is  high,  that  the  rings 
cannot  be  seen  in  the  instruments,  the  rays  following  the  optic 
axes  being  totally  reflected  in  air.  In  such  cases  the  angle 
can  be  measured  by  the  deviation  of  the  refracted  rays, 
diminished  by  observing  their  emergence  in  a  denser  medium 
than  air.  This  is  done  by  placing  a  parallel-sided  glass 
cistern,  filled  with  a  colourless  oil  whose  refractive  index  is 
known,  in  the  space  between  the  condenser  and  objective  of 
the  instrument,  into  which  the  crystal  plate  is  immersed  ;  the 
angle  between  the  axes  is  then  determined  by  turning  the 
plate  about  its  centre,  and  reading  the  goniometer  as  before. 
This  gives  the  apparent  angle  in  oil,  2  H,  and  from  it  the 
refractive  index  of  the  o^=  «,  and  ft,  the  true  angle  is  found 
by  the  formula : 

sin^F=  |  sin.  H. 

The  angle  of  the  optic  axes  may  also  be  calculated  with- 
out knowing  the  refractive  indices  ft  and  n,  if  a  second  plate 
can  be  obtained  normal  to  their  obtuse  bisectrix  or  second 
median  line,  with  which  the  obtuse  angle  is  measured  in 
oil.  Calling  this  latter  2J7',  and  the  corresponding  true 
angle  2  V,  their  relation  will  be  expressed,  as  in  the  pre- 
ceding case,  by — 


CHAP.  XII.]      Calculation  of  Optic  Axial  Angles.       265 

sin.  V  =  2  sin.  H', 
P 

which  requires  a  knowledge  of  the  refractive  indices.  But 
as  the  sum  of  the  acute  and  obtuse  angles  for  the  same 
coloured  light  is  always  180°,  V+  V  =  90°  and  sin. 
V  =  cos.  V.  Substituting  this  value  in  the  preceding 
equation,  it  becomes  : 

cos.  F=  ^  sin.  H1, 
which,  divided  into  the  formula  for  the  acute  angle, 

sin.  r=|sin.J5r, 

gives 

tan.  r=ss|n-g;, 

which  expression  determines  the  true  angle  of  the  optic  axis 
without  the  use  of  any  refractive  indices  ;  and  if  the  acute 
angle  can  be  measured  in  air,  as  the  true  angle  is  known,  the 
mean  refractive  index  can  also  be  found,  because, 

n sin.  V 

~~  sin.  E' 

This  method  is  one  of  considerable  practical  value,  as  it  often 
allows  the  determination  of  the  principal  optical  constants 
of  a  mineral  to  be  made  when  the  crystals  are  too  small  to 
furnish  more  than  one  prism  of  a^lufficiently  accurate  form 
for  direct  observation,  a  very  mimMWragment  of  a  parallel- 
sided  plate,  which  need  not  be  cut  so  exactly  true  for  direc- 
tion as  the  prism,  being  sufficient  to  show  the  interference 
figures  in  convergent  polarised  light. 

The  angle  of  the  optic  axes  found  by  the  above  method 
is  only  true  for  the  particular  colour  of  the  light  used,  and  it 
will  have  some  other  value,  either  greater  or  less,  for  the 
other  colours  of  the  spectrum.  The  amount  of  the  dif- 
ference, which  is  known  as  the  dispersion  of  the  optic  axes, 
is  specific  for  any  particular  mineral,  being  sometimes  only 


266  Systematic  Mineralogy.  [CHAP.  XII. 

a  few  minutes,  while  at  others  it  may  be  from  thirty  to  forty 
degrees  or  more.  Furthermore,  the  dispersion  may  take 
place  in  several  different  ways,  each  being  characterised 
by  a  particular  effect  on  the  interference  figures  in  white 
light. 

In  the  rhombic  system  the  median  line  is  common  for 
light  of  all  colours,  the  axes  being  dispersed  in  the  same 
plane  symmetrically  to  it,  and  therefore  the  colours  of  the 
interference  rings  are  arranged  symmetrically  to  both  arms 
of  the  cross  when  the  plane  of  the  optic  axis  is  parallel  to 
that  of  one  of  the  crossed  Nicols,  but  the  colours  of  the  in- 
dividual rings  vary  with  the  dispersion.  If  the  angle  for  red 
light  be  less  than  that  for  violet,  which  is  indicated  by  the 
symbol  p<v,  the  field  enclosed  by  the  polar  rings  about 
either  axis  will  show  a  red  margin  on  the  inner  and  a  blue 
one  on  the  outer  side,  as  in  fig.  357,  the  colour  being  exactly 
similarly  distributed  on  both  sides  of  the  horizontal  line,  and 
when  the  plate  is  turned  through  45  degrees,  the  brushes  will 

FIG.  357.  FIG.  358. 


be  bordered  with  blue  on  their  inner  or  convex  sides,  and 
red  on  the  outside,  the  colour  within  the  rays  remaining  un- 
altered as  in  fig.  35 8. l  This  class  of  dispersion  is  charac- 
teristic of  Nitre ;  while  the  opposite  condition,  where  o  >  v 
and  the  hyperbolas  are  bordered  with  red  within  and  blue 
without,  is  seen  in  the  allied  minerals  Aragonite  and  White 

1  The  reader  is  recommended  to  colour  these  and  the  succeeding 
diagrams,  when  the  differences  will  be  more  readily  appreciated. 


CHAP.  XII.]  Rhombic  Dispersion.  267 

Lead  Ore.  In  the  latter,  owing  to  the  high  specific  refrac- 
tive and  dispersive  power,  the  phenomenon  is  very  strikingly 
shown,  the  brushes  appearing  as  broad  red  and  blue  stripes, 
without  an  intervening  dark  space. 

In  a  small  number  of  crystallised  substances  belonging 
to  the  rhombic  system,  of  which  Brookite  is  the  principal 
natural  example,  the  axes  of  differ-  FIG.  359. 

ent   colours,  though  having  the 
same  median  line,  are  not  only 
widely     dispersed,    but    lie    in 
different    planes,   those    for  redj 
being  in   the  plane  indicated  by  '^ 
one  bar  of  the  cross,  while  those 
for  green  are  in  the  other  bar  at 
right  angles  to  it.     In  white  light 
these  crystals  give  very  peculiar 

interference  figures,  like  fig.  359,  the  rings  being  replaced 
by  a  series  of  curves  symmetrical  to  the  cross  upon  a  parti- 
coloured ground,  the  maximum  of  red  being  about  the 
horizontal  bar  of  the  cross,  and  that  of  green  near  the 
vertical  one.  If,  however,  this  is  viewed  in  homogeneous 
red  light,  it  is  resolved  into  a  series  of  lengthened  oval 
rings,  whose  poles  are  at  r,  r1,  while  with  green  light 
another  series,  more  nearly  circular  in  form,  are  seen,  having 
their  poles  at  r,  r1.  The  same  phenomena  are  perhaps  more 
strikingly  seen  in  the  triple  Tartrate  of  Potassium,  Sodium, 
and  Ammonium,  known  as  Sel  de  Seignette,  which  is  not 
only  more  readily  obtained  than  a  crystal  of  Brookite,  but, 
being  colourless,  shows  the  colours  of  the.  field  more  vividly. 
With  a  section  of  this  salt  the  red  and  blue  ring-systems 
may  be  seen  superposed,  when  the  middle  part  of  the  spec- 
trum is  extinguished  by  a  tolerably  thick  cobalt  blue 
glass. 

In  the  oblique  system,  the  optic  axes  corresponding  to 
different  colours  have  not  necessarily  a  common  median 
line,  and  therefore  a  new  element  is  presented  for  conside- 


268  Systematic  Mineralogy.          [CHAP,  xn. 

ration,  namely,  the  dispersion  of  the  median  lines.  This 
may  take  place  in  three  different  ways,  each  having  a  more 
or  less  characteristic  effect  upon  the  interference  figures, 
and  to  these  the  names  inclined,  horizontal,  and  crossed 
dispersion  have  been  applied  by  Descloizeaux,  who  first 
systematically  investigated  them. 

The  first  case,  that  of  inclined  dispersion,  arises  when 
the  optic  axes  lie  in  the  plane  of  symmetry  dispersed  at  in- 
clinations which  are  usually  small,  their  median  lines  making 
angles  with  each  other  varying  from  a  few  minutes  to  one  or 
two  degrees.  Of  the  latter,  therefore,  only  one,  usually  that 
for  the  middle  of  the  spectrum,  can  coincide  with  the  normal 
to  the  plate,  and  consequently  those  for  the  other  colours 
will  be  unsymmetrically  placed  right  and  left  of  the  centre. 
When  such  a  crystal  is  placed  with  its  axial  plane  parallel 
to  that  of  one  of  the  Nicols,  there  will  be  often  seen  a 
marked  difference  in  the  shape  and  size  of  the  polar  rings, 
one  being  nearly  circular,  while  the  other  is  a  lengthened 

FIG.  360.  FIG.  361. 


ellipse,  as  in  fig.  360,  the  vertical  bar  of  the  cross  is  nearer 
to  the  latter  than  the  former,  and  the  colours  are  generally 
brighter  about  one  pole  than  the  other,  the  same  order  being 
observed  when  the  plate  is  turned  through  45°,  as  in  fig.  361 ; 
but  the  colours  of  the  brushes  in  the  latter  position  may  be 
either  opposed,  one  being  blue  inside  and  red  outside,  and 
the  other  blue  outside  and  red  inside,  or  vice  versa,  or  similar, 
the  phenomena  being  complicated  by  differences  in  the  dis- 


CHAP.  XII.]  Oblique  Dispersion.  269 

persion  of  the  optic  axes,  as  well  as  by  that  of  their  median 
lines.  This  kind  of  dispersion  is  best  seen  in  Diopside, 
Gypsum  and  other  minerals  having  large  angles  between 
their  optic  axes ;  the  contrast  between  the  form  of  the  polar 
rings  is  seen  in  the  artificial  salt,  Platino- Cyanide  of  Barium. 
When  no  polar  rings  are  apparent,  as  in  the  small  angled 
Felspars,  there  is  only  a  slight  difference  of  brilliancy  of 
the  colours  of  the  brushes,  which  is  not  clearly  appreciable 
without  practice  in  observing. 

When  the  plane  of  the  optic  axis  is  perpendicular  to  the 
plane  of  symmetry  and  parallel  to  the  orthodiagonal,  the 
latter  may  be  either  perpendicular  to  the  first  meridian 
line,  or  parallel  to  it.  The  former  position  corresponds  to 
the  second  case,  or  that  of  horizontal  dispersion,  the  axes 
corresponding  to  one  colour  only,  being  situated  in  a  plane 
parallel  to  the  orthodiagonal,  while  those  for  other  colours 
are  in  planes  making  a  slightly  different  angle  with  the  clino- 
diagonal  axis,  but  all  their  meridian  lines  lie  in  the  plane  of 
symmetry,  so  that  the  horizontal  bars  of  the  cross,  in  the  in- 
terference figures  for  the  extreme  colours  will  not  lie  in  the 
same  line,  but  will  be  parallel  to  each  other  with  more  or 
less  horizontal  displacement.  If  therefore  the  plate  is  cut 
true  for  a  mean  colour,  as  yellow  or  green,  the  horizontal 
bar  in  the  position  of  greatest  obscuration  will  be  bordered 

FIG.  362.  FIG.  363. 


with  red  on  one  side  and  blue  on  the  other,  as  in  fig.  362, 
the  corresponding  colours  of  the  brushes  in  the  diagonal 
position  being  seen  in  fig.  363. 


2/o  Systematic  Mineralogy.          [CHAP.  XII. 

In  the  third  case,  that  of  crossed  dispersion,  the  median 
lines  for  all  colours  coincide  in  direction  with  the  ortho dia- 
gonal, but  the  planes  of  their  optic  axes  cross  at  small  angles. 
The  colours  of  the  rings  and  brushes  are  therefore  disposed 
chequerwise,  as  in  fig.  364,  the  maximum  of  red  in  the  left 

FIG.  364.  FIG.  365. 


polar  field  being  above  the  horizontal  bar  and  to  the  left, 
and  that  of  blue,  below  and  to  the  right ;  while  in  the  left 
one  the  positions  are  reversed,  a  similar  contrasted  arrange- 
ment is  also  apparent  in  the  diagonal  position,  fig.  365. 
This  kind  of  dispersion  is  well  seen  in  Borax  crystals,  and 
to  a  lesser  degree  in  Gay  Lussite. 

In  the  triclinic  system  no  direct  relation  is  apparent 
between  the  crystalline  form  and  the  interference  figures, 
two  or  more  kinds  of  dispersion  being  sometimes  seen  in 
the  same  crystal. 

Determination  of  the  sign  of  the  double  refraction.  The 
positive  or  negative  character  of  a  doubly  refracting  mineral 
may  be  determined  in  most  cases  from  a  section  showing  its 
interference  rings,  by  placing  below  the  analyser  a  plate  of 
another  mineral  Avhose  sign  is  known,  and  observing  the 
effect  upon  the  figure.  With  uniaxial  crystals  the  simplest 
method  that  can  be  used  is  to  place  above  the  section  under 
examination  a  plate  of  some  other  uniaxial  crystal,  when,  if 
both  have  the  same  sign,  they  will  act  together,  producing 
the  effect  of  an  apparent  thickening  of  the  lower  plate,  and 
the  interference  rings  will  appear  closer  together ;  but  if  they 
are  of  dissimilar  signs,  one  will  partially  neutralise  the  effect 
of  the  other,  and  the  rings  of  the  first  will  be  expanded  as 


CHAP.  XII.]  Determination  of  Sign.  271 

though  its  thickness  had  been  reduced.  This  method  is, 
however,  but  of  very  limited  application. 

When  a  section  of  a  uniaxial  crystal,  parallel  or  but 
slightly  inclined  to  the  optic  axis,  is  examined  in  convergent 
polarised  light  between  crossed  Nicols,  no  coloured  rings  are 
observed  except  it  be  very  thin,  when  a  series  of  hyperbolic 
bands  symmetrical  to  a  central  cross  appear.  If,  therefore, 
a  plate  of  sufficient  thickness  be  placed  so  as  to  give  a  field 
of  maximum  brightness,  in  which  position  its  planes  of  vibra- 
tion are  at  45°  to  those  of  the  Nicols,  and  a  tapered  wedge 
of  Quartz,  whose  length  is  parallel  to  its  optic  axes,  be 
passed  below  the  analyser,  first  in  the  direction  of  one  vibra- 
tion plane  and  then  of  the  other,  the  hyperbolic  bands  will 
be  seen  in  one  or  other  position.  The  reason  of  this  is,  that 
Quartz  being  positive,  its  optic  axis  coincides  with  that  of 
its  minimum  elasticity,  and  the  particular  direction  in  which 
it  is  without  effect  or  appears  to  augment  the  thickness  of 
the  plate  is  the  axis  of  like  elasticity  in  the  latter,  while  the 
other  direction  at  right  angles  to  the  first  is  that  of  maximum 
elasticity  in  the  plate,  and  opposed  to  that  of  the  Quartz, 
which  therefore  acts  as  though  the  plate  were  thinned.  If, 
therefore,  the  latter  direction  is  that  of  the  optic  axis,  the 
double  refraction  of  the  crystal  is  opposed  to  that  of  the 
Quartz,  or  is  negative,  while  in  the  other  case  it  is  similar,  or 
positive.  The  particular  thickness  required  for  this  so-called 
compensating  action  varies  with  that  of  the  plate,  and  there- 
fore a  certain  length  of  Quartz  is  required  to  give  the  neces- 
sary range.  As  ordinarily  made,  the  wedge  is  about  if  inches 
long  and  TV  inch  at  the  thickest  end. 

The  same  method  is  applied  with  sections  of  biaxial 
crystals,  showing  the  interference  rings  and  brushes  by  in- 
serting the  Quartz  wedge,  first  in  the  direction  of  the  line 
joining  the  optic  axis,  and  then  at  right  angles  to  it.  If  the 
rings  appear  to  expand  in  the  first  position  the  crystal  is 
positive  ;  but  if  they  are  unchanged  or  completely  effaced, 
it  is  negative,  and  the  expansion  or  production  of  the  hyper- 
bolic bands  will  take  place  at  right  angles  to  the  axial  plane. 


272  Systematic  Mineralogy.          [CHAP.  XII. 

A  third  method  which  is  specially  applicable  to  uniaxial 
crystals  and  to  biaxial  ones  in  sections  that  are  too  thin  to 
show  the  polar  rings,  depends  upon  the  use  of  circularly 
polarised  light,  either  in  the  polariser,  or  the  analyser.  If  a 
plate  of  biaxial  Mica  sufficiently  thin  to  produce  a  retarda- 
tion of  phase  of  exactly  a  quarter  wave  length  between  the 
refracted  rays  be  placed  in  the  path  of  a  plane  polarised 
beam,  with  the  plane  of  its  optic  axes  that  of  45  degrees  to 
the  polariser,  the  light  will  emerge  in  a  condition  of  circular 
polarisation,  and  the  dark  line  marking  the  plane  of  vibra- 
tion of  the  polariser  will  no  longer  be  apparent. }  A  section 
of  a  uniaxial  crystal  viewed  under  these  conditions  no  longer 
presents  the  concentric  rings  and  dark  cross  between  curved 
Nicols,  but  that  shown  in  figs.  366-7,  the  rings  being  broken 

FIG.  367. 


into  four  parts,which,when  compared  with  their  original  forms, 
are  expanded  and  contracted  in  alternate  quadrants.  The 
contracted  parts  having  the  greatest  intensity  of  obscuration, 
those  of  the  innermost  ring  will  appear  as  two  dark  or 
coloured  spots,  and  a  line  joining  them  will  be  either 
parallel  or  perpendicular  to  the  axial  plane  of  the  Mica 
plate.  If,  therefore,  the  latter  is  always  used  in  one  posi- 
tion, with  its  axial  line  diagonal  to  the  first  and  third  quad- 
rants of  the  circle  if  the  spots  lie  upon  it,  as  in  fig.  366, 
the  crystal  is  negative  ;  but  when  they  are  in  the  second  and 
fourth  quadrants,  or  at  right  angles  to  it,  as  in  fig.  36^  the 
crystal  is  positive.  The  same  thing  is  observed  with  braxial 

1  An  explanation  of  this  change  of  polarisation  will  be  found  in 
Spottiswoode's  treatise  on  polarised  light,  p.  50. 


CHAP.  XII.] 


Determination  of  Sign. 


273 


crystals  of  small  angle,  but  in  those  of  wide  angles  the  rings 
are  expanded  above  and  below  the  horizontal  line,  so  that 
if  the  field  be  supposed  to  be  divided  into  four  quadrants  as 
before,  a  negative  crystal  will  show  a  spot  above  the  line  in 
the  first,  and  below  it  in  the  third  quadrant  (fig.  368)  ;  while 


FIG.  36 


FIG.  369. 


TIT 


IS 


in  a  positive  one  the  spots  will  be  below  the  line  in  the 
second,  and  above  it  in  the  fourth  quadrant,  as  in  fig.  369. 

If  the  light  be  circularly  analysed  as  well  as  polarised, 
by  the  use  of  a  second  quarter  undulation  plate  below  the 
analyser,  the  interference  figures  will  appear  as  complete 
rings  withouj  any  dark  cross,  and  will  behave  in  the  same 
way  as  those  observed  in  a  circularly  polarising  crystal,  the 
rings  expanding  or  contracting  as  the  analyser  is  turned  one 
way  or  the  other.  If  the  polariser  and  analyser  be  placed 
parallel  with  the  axial  plane  of  the  plate  under  examination 
in  the  vertical  line  of  the  field,  and  the  two  quarter  undula- 
tion plates  with  their  axial  planes  crossed  at  right  angles  or 
at  45°  to  that  of  the  plate,  the  latter  will,  if  negative, 
behave  as  a  right-handed  crystal,  or  the  rings  will  expand 
when  the  analyser  is  turned  to  the  right ;  but  if  positive,  the 
rotation  must  be  to  the  left  to  produce  the  same  effect. 
This  is  one  of  the  best  methods  for  use  with  crystals  whose 
interference  rings  are  small  and  close  together. 

Irregular  polarisation.  In  some  cases  crystals  belonging 
to  the  cubic  system,  when  examined  under  polarised  light, 
exhibit  traces  of  double  refraction  not  compatible  with  the 
assumption  of  uniform  structure  required  by  the  system. 
The  most  notable  examples  are  Alum,  Boracite,  and  Senar- 
montite.  These  have  been  variously  explained ;  as  for 


274  Systematic  Mineralogy.          [CHAP.  XII. 

example,  in  Alum,  by  the  assumption  of  lamellar  structure 
due  to  the  successive  layers  of  the  crystal  not  being  in  abso- 
lute contact,  and  therefore  capable  of  polarising  light  in 
the  same  way  as  a  bundle  of  glass  plates,  or  by  the  existence 
of  strains  in  the  interior  of  the  crystal,  producing  a  structure 
analogous  to  that  of  unannealed  glass,  and  in  Boracite  by 
the  symmetrical  inclusion  of  doubly  refracting  substances  in 
very  minute  crystals.  The  polarising  character  of  such  mine- 
rals is  not  constant,  differing  in  different  parts  of  the  same 
section,  and  being  quite  independent  of  any  particular  direc- 
tion of  the  crystal.  Similar  irregularities  are  often  observed 
in  uniaxial  crystals,  the  interference  rings  and  cross  behaving 
like  those  of  biaxial  ones  of  small  angle.  Here  again,  Tiow- 
ever,  the  disturbance  is  commonly  local,  and  a  part  of  the 
plate  may  often  be  found  giving  the  proper  figure,  and  in 
the  disturbed  figure  the  innermost  ring  is  not  a  continuous 
curve  as  that  of  a  truly  biaxial  crystal  should  be,  but  is  made 
up  of  disconnected  portions  of  circles  of  different  radii. 

A  general  explanation  of  the  anomalous  optical  behaviour 
of  minerals  applicable  to  all  the  known  cases  has  been  put 
forward  at  great  length  in  an  admirable  memoir  by  Professor 
Mallard,1  who  supposes  these  phenomena  to  be  indications 
of  polysynthetic  structure,  simple  crystals,  either  asymmetric 
or  of  systems  of  low  symmetry,  by  many  repetitions  produc- 
ing groups  which  simulate  simple  forms  of  more  complex 
symmetry. 

The  presence  of  foreign  substances  in  transparent 
minerals,  as  well  as  of  twinned  structure,  is  often  very  strik- 
ingly shown  in  parallel  polarised  light.  Fig.  370  is  an  ex- 
ample of  a  section  of  an  apparently  homogeneous  Quartz 
crystal  perpendicular  to  the  optic  axis,  in  which  all  the  light 
parts  have  one  rotation  and  the  shaded  ones  the  opposite, 
thus  proving  it  to  be  a  really  complex  twin  of  many  right- 
and  left-handed  individuals  with  irregular  contact  surfaces. 

1  Annales  des  Alines,  3  ser.  vol.  x.  p.  60. 


CHAP.  XII.]  Irregular  Structure.  275 

Fig.  371  is  a  section  of  a  twinned  group  of  Aragonite,  in 
which  the  planes  of  the  optic  axes  in  the  alternate  indi- 
viduals are  inclined  to  each  other  as  shown  by  the  lines  and 
rings.  If  parts  of  the  two  adjacent  bands  be  seen  together 

FIG.  370.  FIG.  371. 


in  convergent  light,  both  systems  of  interference  rings,  crossed 
in  direction,  will  often  be  apparent  at  the  same  time,  but  by 
shifting  the  plate  slightly  either  to  right  or  left  one  of  them 
will  disappear.  The  same  thing  is  seen  in  Carbonate  of 
Lead  and  Strontianite,  minerals  whose  crystals  are  analogous 
in  structure. 

The  subject  of  minute  inclosures  of  foreign  minerals,  as 
well  as  that  of  the  preparation  of  thin  sections  suitable  for 
optical  examination,  has  already  been  treated  at  length  in 
the  treatise  on  Rocks  in  this  series,  to  which  the  student  is 
referred.  It  may,  however,  be  useful  to  remark  that  the 
sections  required  for  showing  interference  rings  are,  as  a 
rule,  much  thicker  than  those  prepared  for  microscopic  in- 
vestigation under  parallel  light.  In  the  greater  number  of 
the  cases  the  most  useful  thicknesses  will  lie  between  £  and 
i1^  of  an  inch  ;  but  the  actual  size  of  the  fragment,  apart 
from  its  thickness,  is  immaterial ;  for  example,  the  whole  of 
the  rings  may  be  seen  and  the  character  of  the  double  refrac- 
tion determined  on  a  plate  of  Mica  of  two  or  three  hun- 
dredths  of  an  inch  on  the  side. 

The  preparation  of  sections  for  the  polariscope  is  much 
facilitated  by  the  existence  of  a  perfect  cleavage  parallel  to 
the  optic  axis  in  uniaxial,  or  to  the  first  median  line  in  bi- 
axial, crystals,  as  with  these  cleavage  plates  will  usually  suffice 


276  Systematic  Mineralogy.  [CHAP.  xii. 

without  specially  grinding  or  polishing.  Among  the  former 
are  the  tetragonal  variety  of  Sulphate  of  Nickel,  which  sepa- 
rates from  a  saturated  solution  at  temperatures  above  15° 
Cent;  the  native  Phosphate  of  Copper  and  Uranium  also 
tetragonal.  The  transparent  crystals  of  Molybdate  of  Lead 
(Wulfenite)  from  Utah,  which  are  tabular  to  the  basal 
pinakoid,  may  be  used  without  any  preparation ;  but  as  the 
specific  refractive  power  is  very  high,  only  the  thinnest  will 
give  a  system  of  rings  of  any  great  width.  Among  biaxial  sub- 
stances the  best  examples  are  the  species  of  the  Mica  group, 
which  are  susceptible  of  almost  unlimited  cleavage,  and  are 
therefore  well  suited  for  illustrating  the  alteration  of  the  rings 
with  the  thickness  of  the  plate.  Topaz,  Sugar  and  Bichro- 
mate of  Potassium  have  perfect  cleavages  normal  to  one  of 
the  optic  axes,  so  that  from  them  plates  may  be  easily  ob- 
tained showing  the  rings  about  one  axis  only. 

The  following  table  shows  the  optical  constants  of  the 
principal  transparent  minerals.  It  is  for  the  most  part  com- 
piled from  that  given  in  the  'Annuaire  du  Bureau  des 
Longitudes,'  and  the  works  of  Descloizeaux  and  Groth. 

[Note  to  page  263.]— The  optical  elements  of  a  biaxial  crystal  are 
related  in  the  following  manner  : 

Axes  of  elasticity  a   :          b      :          c 

Refractive  indices  (min.)  a  :  (mean)  /3  :  (max.)  j 

Velocities  L  :          1:1 

a  J3  y 

Coefficients  of  elasticity  : 

a"  /32  7* 

The  true  angle  of  the  optic  axes  with  the  median  line  is  found, 
when  the  three  refractive  indices  are  known,  by  the  formula — 


CHAP.  XII.] 


Optical  Constants. 


277 


OPTICAL  CONSTANTS  OF  THE  PRINCIPAL 
TRANSPARENT  MINERALS. 

ISOTROPIC. 


Amorphous  and  Cubic 

Ray 

Refractive 
index 

Water     .         .         .         .         .         .        ..  . 

_ 

I  '336 

Hydrophane,  dry      ..... 

red 

•387 

Do.  saturated  with  water       .         .      :  . 

'439 

Hyalite   ....... 

'437 

Opal,  iridescent  variety    .... 

•446 

Quartz,  melted          ..... 

'457 

Fluorspar,  green       .         .         .         .         . 

'433 

Alum       ....... 

•458 

Sylvine             ...... 

yellow 

•482 

Analcime         ...... 

red 

•487 

Plate  glass  (mean)            ,.         .  ' 

— 

•530 

Agate,  light  coloured        .         .        . 

» 

•537 

Rock  Salt         ...."'. 

yellow 

'543 

•642 

Boracite  ....... 

•667 

Spinal  (rose  colour)  ..... 

red 

•712 

Arsenious  Acid         ..... 

•748 

'772 

,,       Cinnamon  Stone   .... 

/  /  ** 

•741 

Senarmontite   .         .               •>   . 

2-073 

Zincblende  (yellow)  ..... 

2-341 

Diamond  (colourless)        .... 

2-414 

,,          (brown)     ..... 

2-487 

Ruby  Copper  Ore    ..... 

2-849 

ANISOTROPIC  UNIAXIAL. 


Tetragonal  positive 

Ray 

Indices 

CO 

£ 

Leucite        •  . 

— 

I-508 

I-509 

Apophyllite   .... 

red 

I-53I7 

I-533I 

Scheelite        .... 

»j 

1-918 

1-934 

Zircon  ..... 

,, 

1-92 

I'97 

Phosgenite     .... 

orange 

2-114 

2-I40 

Anatase          .         .  •       . 

2-554 

2-493 

278 


Systematic  Mineralogy.          [CHAP.  XII. 


Tetragonal  negative 

Ray 

Indices 

6 

CD 

Mellite  

yellow 

I-525 

I'550 

Meionite        .... 

,, 

I-560 

1-595 

Melinophane 

red 

I-592 

1-611 

Idocrase         .... 

yellow 

I-7I7 

1-719 

Wulfenite      .... 

red 

2-304 

2-402 

HEXAGONAL. 


Positive 

Ray 

Indices 

H 

€ 

Ice  (mean  index)    . 

yellow 

•309 



Quartz  ..... 

» 

'544 

*  '553 

Parisite          .... 

red 

•569 

1-670 

Phenakite      .... 

»> 

•654 

1-670 

Dioptase        .... 

green 

•667 

1-723 

Greenockite  .... 

yellow 

2-688 

Cinnabar       .... 

red 

2-816 

3-142 

Negative 

Ray 

Indices 

E 

ca 

Nitrate  of  Sodium  . 

yellow 

I-336 

•586 

Hedyphane    .... 

red 

I-463 

•467 

Pyromorphite 

,, 

I-465 

"474 

Calcite  

yellow 

I-486 

•659 

Apatite.         .... 

— 



•657 

Dolomite       .... 

H 

I-503 

•612 

Nepheline      .... 

>i 

1-537 

•542 

Pennine         .... 

red 

1-576 

'577 

Emerald         .... 

green 

1-573 

•584 

Tourmaline  (green) 

red 

1-620 

•641 

Corundum  (Ruby)  . 

H 

i  '759 

•767 

Ruby  Silver  Ore  (dark)  . 

II 

2-881 

3-084 

Do.  light  (Proustite)  . 

yellow 

2-792 

3-088 

CHAP.  XII.] 


Optical  Constants. 


279 


ANISOTROPIC  BIAXIAL. 


Indices 

Angle  of 

optic  axes 

Disper- 

Ray 

Min.     Mean 

Max. 

Real 

2V 

Apparen 

2  E 

sion 

RHOMBIC-  POSITIVE  — 
Thenardite          .        .    red 

—          1*470 

o     / 

°_' 

Natrolite    .        .              ,, 

i  '477 

1-480 

1-489 

59  '29 

94*27 

Struvite                               „ 

— 

1  '497 

— 

Harmotome        .               „ 



1*516 



Anhydrite                           „ 

i'57i 

1*614 

— 

43*30 

Electric  Calamine      .  yellow 

1*614 

1*617 

1-636 

46*09 

78*39 

P  •>  v 

Topaz  (white)      .               „ 

I"6l2 

1*615 

1*622 

56*39 

ioo'4o 

P  >  v 

Celestine     .        .        .      „ 

— 

1*625 

— 

— 

89*36 

Barytes       .         .         .      „ 

1*636 

1*637 

1*648 

— 

63*12 

P  <  v 

Olivine  (Peridote)      .     red 

1*661 

1.678 

1*697 

87-46 

p  <  v 

Zoisite                                  „ 



1*700 

98-2 

Diaspore    .        .        .  yellow 

— 

1*722 



Chrysoberyl        .               ,, 
Staurolite   .        .        .     red 

i  '747 

1*748 

1-757 

— 

42  '50 

p  >  v 
p  >  v 

Anglesite    .        .        .      „         1*874     1*880 

1*892 

66*40 

89*49 

P  <   v 

Sulphur       .         .         .  yellow 
-NEGATIVE— 

1*958      2*038 

2*240 

69*40 

70°  to  75° 

p   <  v 

Sulphate  of  Sodium   .      ,, 

— 

1*440 



Sulphate    of   Magne- 

sium                              ,, 

1*4325    1*4554 

1*4608 

— 

77  '5° 

Sulphate  of  Zinc         .      „ 

1*457  .   1*480 

1*484 

— 

70*16 

Nitre  .         .         .         .      „ 

i  '333  i  1*5046 

7*12 

— 

Andalusite  .         .         .     red 

i  '632      i  '638 

i  '643 

3°'I4 

p    <    V 

Autunite     .        .        .      ,, 

— 

I  '572 

Cordierite  .         .         .  orange 

i  '562      1*561 

1*563 

Carbonate  of  Lead     .  yellow 

1*804      2*076 

2*078 

8*07 

16*54 

JBLIQUE-POSITIVE  — 

Ferrous  Sulphate       .      „ 

1*471      1*478 

1-486 

— 

85-27 

Gypsum                              ,, 

1*521      1*527 

i'53° 

— 

61*24 

inclined 

Euclase       .                        ,, 

1*652      1*655 

1-671 

87*59 

inclined 

Anthophyllite     .         .     red 

—        1-636 

— 

81-05 

p  >  v 

Diopside     .        .        .  yellow 

1.673 

1*679 

i'7°3 

58*59 

in'34 

inclined 

Sphene        .         .         .     red 

1*903 

— 

53  '30 

p    >    V 

-NEGATIVE  — 

Borax          .         .        .  yellow 
Adularia  S.  Gotthard.      ,, 

1*447  1  I-469 
1-519  i   1*524 

1-472 
1*526 

39*36 
69*43 

59  '23 
121*06 

crossed 
orizontal 

„         Eifel    .         .     red 

i  '5240 

J3'34 

20*45 

inclined 

Muscovite  (ural)         .  yellow 

i'54i 

I-574 

40'2I 

64*14 

p  >      V 

Treraolite   .                        ,, 

1*622 

87*31 

inclined 

Aclinolite                           „ 

— 

1*629 

. 

80*04 

— 

inclined 

Epidote       .        .        .     red 

i  '731 

1*754 

1-761 

73'36 

— 

inclined 

Malachite   .        .        .     — 

1*88 

— 

43  '54 

89*18 

inclined 

TRICLINIC-POSITIVE  — 

Sillimanite                          „ 

— 

1*66 



44*0 

.NEGATIVE  — 

Sulphate  of  Copper    .  yellow 

1*516 

i'539 

1*546 

— 

96° 

p  <  v 

Axinite       .        .        .     red 

1*672 

1*678 

1*681 

7i  '38 

158-13 

p  <  v 

Amblygonite  (Monte- 

bras)                               ,, 

— 

i  '592 

— 

Cyanite                              „ 

1-720 

° 

280  Systematic  Mineralogy.         [CHAI-.  XIII. 


CHAPTER  XIII. 

OPTICAL    PROPERTIES    OF   MINERALS — Continued. 

Translucency.  In  systematic  mineralogy,  minerals  are  clas- 
sified as  transparent,  semi-transparent,  translucent  in  different 
degrees,  and  opaque,  according  to  their  power  of  trans- 
mitting ordinary  light  through  their  mass ;  these  terms  being 
used  in  the  popular  sense,  without  reference  to  the  homo- 
geneity or  colour  of  the  substance.  The  test  of  transparency 
is  the  power  of  discerning  an  object  through  a  parallel-sided 
plate  or  crystal  of  a  certain  thickness.  Rock-crystal,  Calcite, 
Gypsum,  and  Barytes,  and  among  the  ores  of  the  heavy 
metals,  Zincblende  in  its  lighter-coloured  varieties,  are 
among  the  most  transparent  substances  known.  When  the 
object  is  only  imperfectly  seen,  the  substance  is  semi-trans- 
parent ;  when  only  a  cloudy  light  like  that  seen  through 
oiled  paper  or  ground  glass  is  transmitted,  it  is  translucent ; 
when  no  light  is  transmitted,  it  is  opaque.  These  terms  are 
to  a  certain  extent  relative,  particularly  in  the  lower  degrees, 
where  the  thickness  of  the  substance  must  be  considered, 
especially  when  it  is  dark  coloured.  Flint  and  Obsidian, 
for  example,  are  said  to  be  translucent  at  the  edges,  or  in 
thin  splinters,  while  in  thicker  masses  they  are  apparently 
opaque.  Ferric  oxide  and  its  hydrates  are  also  fairly  trans- 
lucent in  minute  microscopic  crystals,  but  opaque  when 
sufficiently  large  to  be  apparent  without  magnifying.  Mag- 
netite, on  the  other  hand,  does  not  appear  to  be  susceptible 
of  transmitting  light  under  any  condition,  and  is  therefore 
opaque,  as  are  also  the  native  metals,  and  most  of  the  heavy 
metallic  sulphides. 

Colour.  When  a  transparent  substance  has  the  power 
of  absorbing  light-rays  of  different  refrangibility  unequally, 
it  will,  when  viewed  in  ordinary  light,  appear  to  be  of  the 
colour  of  the  light  of  greatest  intensity  transmitted.  This 


CHAP.  XIII.]  Colour.  281 

may  be  a  single  colour  or  a  mixture,  its  true  nature  being 
easily  determined  by  examination  with  a  simple  spectro- 
scope. In  describing  minerals,  however,  only  the  apparent 
colour,  as  seen  by  the  unaided  eye,  is  taken  into  account ; 
such  as  are  transparent  and  without  selective  absorption  in 
white  light  being  said  to  be  colourless,  while  others  are  classi- 
fied according  to  a  scale  laid  down  by  Werner,  which  has 
been  adopted  by  mineralogists  in  all  countries,  and  is  one 
of  the  few  instances  in  which  a  uniform  terminology  has 
been  obtained.  It  is  founded  upon  the  use  of  familiar 
coloured  objects  as  standards  of  reference,  distinction  being 
made  between  metallic  and  non-metallic  colours  as  fol- 
lows : — 

METALLIC  COLOURS. 

Copper  red.     As  in  metallic  Copper  and  Red  Nickel  Ore. 

Bronze  red.  Slightly  tarnished  Bronze  and  Magnetic 
Pyrites. 

Bronze  yellow.  Perfectly  fresh  Bronze,  and  newly  frac- 
tured Magnetic  Pyrites. 

Brass  yellow.     Freshly  fractured  Copper  Pyrites. 

Golden  yellow.     Pure  unalloyed  Gold. 

Silver  white  and  Tin  white.  These  are  used  in  a  con- 
ventional sense  for  any  brilliant  opaque  mineral  without 
any  strongly  marked  colour. 

Lead  grey.     Galena,  Antimony  Glance. 

Steel  grey.     Platinum,  Fahlerz. 

Iron  black.     Magnetite,  Graphite. 

\ 
NON-METALLIC  COLOURS. 

There  are  eight  of  these — namely,  white,  grey,  black, 
blue,  green,  yellow,  red,  and  brown,  each  being  divided 
into  numerous  tints  or  varieties,  in  the  following  order,  the 
purest  or  most  characteristic  tint  being  placed  first. 

Whites.  Snow-white,  yellowish-white,  reddish-white, 
greenish-white,  bluish-  or  milk-white,  greyish-white. 


282  Systematic  Mineralogy.        [CHAP.  XIII. 

Greys.  Ash-grey,  bluish-grey,  greenish -grey,  yellowish- 
grey,  reddish-grey,  smoke-grey,  blackish-grey. 

Blacks.  Velvet-black,  greyish-black,  brownish-  or  pitch- 
black,  reddish-black,  greenish-  or  raven-black,  bluish-black. 

Blues.  Prussian-blue,  blackish-blue,  azure,  violet,  lav- 
ender, smalt,  indigo-blue,  sky-blue. 

Greens.  Emerald-green,  grass-green,  verdegris,  celadon, 
mountain-green,  leek-green,  apple-green,  pistachio-green, 
blackish-green,  olive-green,  asparagus-green,  oil-green,  sis- 
kin-green. 

Yellows.  Lemon -yellow,  sulphur,  straw-yellow,  wax- 
yellow,  honey-yellow,  ochre-yellow,  wine-yellow,  Isabella- 
yellow,  orange-yellow. 

Reds.  Carmine,  aurora  or  fire-red,  hyacinth-red,  brick- 
red,  scarlet,  blood-red,  flesh-red,  cochineal-red,  rose-red, 
crimson,  peach-blossom  red,  colombine-red,  cherry-red, 
brownish-red. 

Browns.  Chestnut-brown,  clove-brown,  hair-brown, 
yellowish-brown,  wood-brown  or  umber,  liver-brown,  black- 
ish-brown. 

According  to  intensity,  colours  are  further  qualified  as 
light  or  dark,  pale  or  deep. 

As  a  distinguishing  character  of  minerals,  colour  is  of 
very  unequal  value;  being  constant,  or  showing  but  slight 
variation  from  a  single  tint  in  particular  species,  such  as  the 
native  metals,  the  crystallised  salts  of  Copper,  and  most 
natural  metallic  sulphides ;  while  in  the  larger  number  of 
so-called  non-metallic  minerals  a  single  species  may,  without 
any  great  variation  of  composition,  be  either  colourless  or 
show  a  considerable  range  of  colours.  In  such  cases,  how- 
ever, a  particular  tint  may  often  be  taken  as  an  indication 
of  partial  replacement  of  one  metallic  constituent  by  another, 
as,  for  example,  the  Silicates  of  Magnesium  and  Calcium, 
which,  when  pure,  are  colourless,  pass  through  various 
shades  of  green  to  nearly  black,  in  proportion  as  Magnesium 
is  replaced  in  part  by  the  analogous  dyad  metal,  Iron. 


CHAP.  XIII.]  Colour.  283 

When  Manganese  is  substituted  in  the  same  way,  as  in 
certain  Micas  and  other  silicates,  they  become  red  or 
purple,  and  so  on  in  many  other  cases. 

A  mineral,  ordinarily  colourless,  may  also,  if  transparent, 
appear  to  be  coloured,  by  reason  of  included  foreign  sub- 
stances. Familiar  examples  of  this  are  afforded  by  the 
numerous  varieties  of  Quartz  :  the  purest,  or  Rock  Crystal, 
being  colourless  and  transparent,  while  Amethyst,  Chryso- 
prase,  Cairngorm,  and  Eisenkiesel,  show  different  colours, 
including  purple,  green,  brown,  black,  and  red,  due  either  to 
minute  traces  of  metallic  elements  in  combination,  or  to  par- 
ticles of  ferric  oxide,  carbon,  or  other  opaque  substances, 
visibly  included.  Such  minerals  are  said  to  be  allochro- 
matiC)  or  adventitiously  coloured,  while  those  whose  colour 
is  uniform  and  due  to  their  own  proper  absorption,  are  self- 
coloured,  or  ideochromatic.  Strictly  speaking,  these  latter 
are  the  only  ones  that  can  be  properly  said  to  be  coloured 
minerals. 

Streak.  Useful  aid  in  the  determination  of  minerals  is 
in  many  cases  afforded  by  comparing  the  colour  of  the  mass 
with  that  of  the  streak  or  powder  produced  by  rubbing  on 
a  file  or  upon  a  piece  of  unglazed  porcelain.  With  trans- 
parent minerals  this  is  generally  of  a  lighter  colour,  and 
with  opaque  ones  darker  than  that  of  the  mass.  For 
instance,  Gold,  Copper  Pyrites,  and  Iron  Pyrites  are  often- 
sensibly  of  the  same  brassy  yellow  tint,  but  the  first  gives  a 
streak  of  its  proper  colour,  while  that  of  Iron  Pyrites  is 
black,  and  that  of  Copper  Pyrites  dark  brown.  Hydrated 
and  Anhydrous  Ferric  Oxide,  or,  as  they  are  commonly 
termed,  brown  and  red  Hematite,  also  differ  sensibly  in 
their  streak,  which  is  brown  in  the  first  and  red  in  the 
second. 

Pleochroism.  The  selective  absorptive  power  producing 
colour  is,  in  those  substances  that  are  isotropic,  constant  in 
any  direction,  so  that  they  will  appear  of  the  same  colour 
whether  in  ordinary  or  polarised  light,  but  in  anisotropic 


284  Systematic  Mineralogy.         [CHAP.  xill. 

ones  it  may  vary  with  the  direction,  so  that  a  crystal  may 
appear  to  be  differently  coloured  by  transmitted  light,  ac- 
cording to  the  direction  in  which  it  is  viewed.  This  property, 
to  which  the  general  term  t&pkochrohm  is  applied,  depends 
upon  the  unequal  absorptive  capacity  of  the  crystal  for 
refracted  rays  vibrating  in  different  planes  in  a  manner 
analogous  to  the  differences  in  optic  elasticity.  The  most 
remarkable  example  is  afforded  by  tourmaline,  which  absorbs 
the  whole  of  the  ordinary  ray,  whose  plane  of  vibration  is 
perpendicular  to  the  optic  axis,  or  the  crystal  is  impermeable 
to  light  in  the  direction  of  that  axis,  while  the  extraordinary 
ray  vibrating  parallel  to  it  passes  freely,  and  in  some  cases 
almost  without  change  of  colour,  in  the  direction  of  the 
lateral  axes,  the  latter  property  being  utilised  as  a  method 
of  obtaining  a  single  plane  polarised  ray  by  double  refrac- 
tion and  absorption  in  the  polariscope  known  as  the  Tour- 
maline tongs.  In  less  extreme  cases,  the  absorption  parallel 
to  the  axis  will  only  be  for  particular  colours,  and  the  light 
transmitted  will  show  the  residual  colour,  while  in  a  perpen- 
dicular direction  the  light  will  be  of  another  colour,  usually 
complementary,  or  nearly  so,  to  the  first.  As  the  difference 
will  be  greatest  in  directions  corresponding  to  maximum 
differences  in  velocity,  a  uniaxial  mineral  may  have  two 
distinct  axial  colours,  or  be  dichroic,  and  a  biaxial  one  tri- 
throic,  a  different  tint  being  apparent  in  the  latter  in  the 
direction  of  each  of  the  three  principal  axes  of  optic  elas- 
ticity, when  the  light  transmitted  parallel  to  either  of  them 
is  examined  separately.  This  may  be  done  by  a  Nicol's 
prism  placed  with  its  shorter  diagonal  parallel  to  the  plane 
of  vibration  of  each  ray  successively.  For  instance,  a 
rhombic  crystal  whose  axes  of  form  and  optic  elasticity 
coincide,  when  viewed  in  the  direction  of  the  vertical  axis, 
or  through  the  base,  shows  the  colours  of  the  rays  trans- 
mitted parallel  to  a  lateral  axis,  when  the  principal  section 
of  the  Nicol's  prism  is  parallel  to  that  axis,  and  those  proper 
to  the  vertical  and  a  lateral  axis  may  be  seen  in  the  same 


CHAP.  XIII.]  Dichroiscope.  285 

way  through  either  of  the  other  pinakoids.  Such  differences 
of  colour,  though  possible  in  all  coloured  doubly-refracting 
minerals,  are  not  always  apparent,  the  property  being  pos- 
sessed in  very  unequal  degrees  by  different  minerals,  and  in 
many  of  them  it  is  extremely  feeble.  In  such  cases  it  may 
often  be  rendered  apparent  when  heightened  by  contrast, 
the  colours  proper  to  the  two  axes  in  the  same  section  being 
seen  side  by  side  at  the  same  time,  which  can  be  done  by 
using  a  rhomb  of  Calcite  or  double-image  prism  instead  of 
a  Nicol's  prism.  The  most  convenient  arrangement  of  this 
kind  is  Haidinger's  Dichroiscope,  fig.  372.  It  consists  of  a 
long  cleavage  rhombohedron  Fig  372 

of  Iceland  Spar,  a,  mounted  in 
a  tube,  having  a  glass  prism 
bb'  of  1 8°  refracting  angle  ce- 
mented to  either  end  which 
parallelise  the  incident  and  emergent  rays  to  the  larger  edges 
of  the  prism.  A  metal  cap  with  a  square  hole,  c,  is  fixed  to 
one  end,  and  a  convex  lens,  d,  is  cemented  to  the  glass 
prism  at  the  opposite  end,  which  brings  the  two  images 
produced  by  double  refraction  in  the  Calcite  exactly  parallel 
to  each  other  when  viewed  through  the  eye-piece  cap  o. 
In  white  light  these  images  will  be  exactly  alike  except  as 
regards  intensity,  but  when  a  pleochroic  mineral  is  placed 
in  front  of  c  they  will  be  differently  coloured,  certain  rays 
being  extinguished  in  the  ordinary  beam,  and  their  comple- 
mentaries  in  the  extraordinary  one. 

Some  few  minerals  are  so  strongly  pleochroic  that  the 
differences  of  tint  are  apparent  to  the  unaided  eye.  lolite 
or  Dichroite,  for  instance,  appears  to  be  dark  sapphire-blue 
in  certain  directions,  and  pale  smoky  grey  or  brown  in  others. 
These,  however,  are  not  pure  axial  colours,  but  are  the 
approximate  complementaries  to  those  rays  that  are  most 
completely  absorbed,  and  for  exac  determination  the  use  of 
polarised  light  is  necessary.  Zircon,  Diaspore,  Hornblende, 
the  darker  coloured  Micas,  certain  varieties  of  Chlorite, 


286  Systematic  Mineralogy.         [CHAP.  XIII. 

Andalusite,  Axinite,  and  Epidote,  are  other  examples  of 
strongly  pleochroic  minerals. 

An  interesting  application  of  the  principle  of  selective 
absorption  consequent  on  direction  is  afforded  by  Dove's 
test  of  the  optical  character  of  the  Mica  groups,  many  species 
of  which  give  apparently  uniaxial  interference  figures  when 
viewed  in  convergent  polarised  light,  owing  to  the  very 
small  difference  between  their  minimum  and  mean  refractive 
indices.  A  polarising  instrument  is  arranged  for  parallel 
light,  and  a  plate  of  unannealed  glass  or  calcite  is  viewed 
by  a  plate  of  mica  used  as  an  analyser.  Supposing  the  latter 
to  be  truly  uniaxial,  its  absorptive  power  will  be  equal  for 
rays  vibrating  in  any  azimuth,  and  therefore  no  figure  will 
be  apparent,  but  if  it  be  biaxial,  however  small  the  angle  of 
its  optic  axis  may  be,  every  ray  entering  it  will  be  divided 
into  two,  whose  planes  of  vibration  will  be  parallel  to  the 
axes  of  minimum  and  mean  elasticity  respectively,  the  one 
making  the  largest  angle  with  the  direction  of  the  plane 
of  the  optic  axes  in  the  mica  will  be  most  completely  ab- 
sorbed, and  the  interference  figure  corresponding  to  the 
other  will  be  rendered  visible,  though  usually  in  very  faint 
colours. 

The  success  of  the  test  depends  on  the  power  of  absorp- 
tion, and  therefore  it  is  only  suited  to  those  micas  that  are 
somewhat  coloured  and  transparent  in  moderately  thick 
plates,  amongst  which,  however,  the  apparently  uniaxial 
varieties  are  not  generally  found.  The  same  remark  also 
applies  to  the  use  of  Tourmaline  plates  as  polarisers,  the 
colourless  varieties,  being  almost  without  absorptive  power, 
are  comparatively  valueless  as  compared  with  the  brown  or 
green  ones. 

Lustre.  Light,  when  reflected  from  the  faces  of  crystals 
or  other  surfaces,  is  partly  returned  in  a  regularly  reflected 
beam,  and  partly  irregularly  reflected  or  dispersed,  the  joint 
effect  of  reflection  and  dispersion  being  to  produce  upon  the 
surface  the  peculiar  appearance  known  as  lustre,  glance,  or 


CHAP.  XIII.]  Optical  Lustre.  287 

brilliancy.  In  the  definition  of  lustre,  which  is  often  very 
useful  in  the  proper  determination  of  minerals,  two  points 
are  considered,  namely,  its  quality  or  kind,  which  is  specific 
and  depends  upon  the  refractive  energy  of  the  substance, 
and  its  intensity  or  degree,  which  varies  in  the  same  sub- 
stance with  the  character  of  the  reflecting  surface. 

The  kinds   of   lustre,   commencing  with    the  highest, 
are  : 

1.  Metallic  lustre.     This  is   the  peculiar  and  brilliant 
appearance  seen  upon  a  perfectly  polished  metal  surface. 
It  is  essentially  characteristic  of  the  native  metals  and  heavy 
metallic  sulphides,  and  of  the  few  dark-coloured  transparent 
ones,  such  as  the  Ruby  Silver  Ores  and  Cinnabar  having 
refractive  indices  above  2*5. 

2.  Adamantine  lustre.     The  typical  example  is  the  Dia- 
mond, but  it  is  also  characteristic  of  transparent  minerals, 
whose  refractive  indices  are  from  i-8  to  2*5,  which  include 
the  natural  Sulphide,  Carbonate,  Chloride,  Tungstate,  and 
Molybdate  of  Lead.     The  heavy  lead  glass,  known  as  flint 
glass  or  crystal,  has  also  an  adamantine  lustre  when  polished. 

3.  Vitreous  lustre,  or  that  of  a  glass  not  containing  Lead, 
is  characteristic  of  most  of  the  transparent  crystals  known  as 
gems,  whose  refractive  indices  are  below  i  '8,  Quartz  or  rock 
crystal  being  a  most  familiar  example.     Some  of  the  minerals 
of  this  class,  when  dark -coloured  or  imperfectly  transparent, 
show  a  resinous  lustre,  as  that  of  boiled  pine  resin  or  colo- 
phonium. 

4.  Fatty  or  greasy  lustre  resembles  that  of  a  freshly  oiled 
reflecting  surface,  and  is  characteristic  of  slightly  transparent 
minerals,  such  as  Serpentine,  Nepheline,  and  Sulphur. 

5.  Nacreous  lustre,  or  that  of  the  Mother- of- Pearl  shell, 
is  a  common  characteristic  of  minerals  having  very  perfect 
cleavages,  and  is  best  seen  upon  cleavage  surfaces,  such  as 
those  of  Gypsum  and  Stilbite. 

6.  Silky  lustre  is  essentially  characteristic  of  imperfectly 
translucent  and  fibrous  aggregates  of  crystals.     The  fibrous 


288  Systematic  Mineralogy.         [CHAP.  XIII. 

variety  of  Gypsum  known  as  Satin  Spar,  and  the  native 
Alums,  are  among  the  best  examples. 

There  being  no  absolute  standard  of  classification  in 
regard  to  lustre,  intermediate  terms  are  often  used  in  de- 
scribing that  of  minerals  which  do  not  exactly  correspond 
to  a  particular  kind  in  the  judgment  of  the  observer.  Thus 
anthracite  is  said  to  be  semi-metallic  in  lustre,  certain 
varieties  of  Carbonate  of  Lead  metallic-adamantine,  &c. 
The  definitions  based  upon  the  refractive  power  of  the  sub- 
stances given  are  those  of  Professor  W.  H.  Miller. 

It  is  not  uncommon  to  find  different  faces  of  the  same 
crystal  different  in  lustre,  and  such  differences  are  often  of 
considerable  value  in  fixing  the  position  of  the  forms.  Thus 
in  Quartz,  of  the  two  rhombohedra  making  up  the  unit 
hexagonal  pyramid,  that  considered  as  the  positive  one  is 
generally  brighter  than  the  other  ;  and  the  so-called  rhombic 
faces,  those  of  the  acute  pyramid  of  the  second  order  2  Pz, 
are  so  much  more  brilliant  than  those  of  the  associated 
forms  that  they  may  be  often  detected  by  the  naked  eye, 
even  when  extremely  small. 

As  regards  degree  or  intensity  of  lustre,  minerals  are  said 
to  be  splendent,  shining,  glistening,  or  glimmering,  as  the 
proportion  of  dispersed  to  reflected  light  increases ;  when 
no  distinct  reflection  is  obtained,  the  substance  is  said  to  be 
dull.  The  terms  are,  however,  even  looser  than  those  de- 
fining the  quality  of  the  lustre. 

Iridescence.  The  appearance  of  a  colour,  either  singly  or 
in  variegated  bands  and  patches  on  their  surfaces,  when 
viewed  under  oblique  incident  light,  is  a  well-marked  cha- 
racter of  many  minerals,  some  of  the  most  striking  examples 
being  furnished  by  the  massive  cleavable  Felspar  of  Labra- 
dor, which  very  generally  shows  patches  of  deep  blue  alter- 
nating with  green  upon  the  brachydiagonal  cleavage  planes, 
and  in  rarer  instances  a  much  more  extended  range  of 
colour,  including  rose-red  and  orange -yellow.  Hypersthene 
also  shows  a  bronze  red  tint  upon  the  same  surfaces.  These 


CHAP.  XIII.]  Iridescence.  289 

appearances  have  been  attributed  by  different  investigators 
either  to  structures  proper  to  the  mineral,  such  as  small 
pores  or  cavities  regularly  arranged,  or  to  the  interference 
phenomena  produced  by  very  minute  crystals  of  Horn- 
blende  or  other  minerals,  interspersed  in  the  same  regular 
manner.  In  other  cases,  minerals  which  are  transparent 
and  colourless,  sparkle  with  a  golden  light  by  the  reflection 
from  interspersed  opaque  crystals,  producing  the  so-called 
avanturine  structure  seen  in  Felspars  and  Quartz,  the  in- 
cluded substances  being  in  the  first  case  scales  of  Hematite 
or  Goethite,  and  in  the  last  flakes  of  a  golden-coloured  Mica. 
The  large  Calcite  crystals  from  Lake  Superior  are  sometimes 
coloured  in  the  same  way  by  small  crystals  of  native  Copper 
disseminated  through  them. 

The  brilliant  iris  of  the  Opal  is  attributed  to  the  presence 
of  minute  faces  crossing  the  mass  in  different  directions,  and 
in  one  variety  known  as  Hydrophane,  the  opalescence  dis- 
appears when  the  substance  is  soaked  in  water. 

In  minerals  with  very  perfect  cleavage  coloured  rings  are 
frequently  seen  at  different  points  in  the  interior.  These 
are  the  ordinary  colours  of  thin  plates  produced  by  minute 
films  of  air  included  between  cleavage  surfaces.  They  may 
be  seen  in  almost  any  large  clear  specimen  of  Mica,  Gypsum, 
or  Calcite. 

Surface  iridescence  or  peacock  colour  is  also  due  to  the 
formation  of  very  thin  layers  of  one  substance  upon  the 
surface  of  another,  and  therefore  marks  the  commencement 
of  alterations  in  the  second.  It  is  usually  spoken  of  as 
iridescent  tarnish,  and  is  seen  in  many  sulphides,  such  as 
Antimony  Glance,  Copper  Pyrites,  and  Purple  Copper  Ore. 
The  Hematite  of  Elba  is  also  remarkable  for  the  brilliant 
rainbow  colouring  often  seen  upon  the  crystals. 

Asterism.  Certain  varieties  of  Corundum  known  as  star 
sapphires,  when  ground  to  a  spherical  form  and  polished, 
show  a  pale  blue  six-rayed  star  when  a  strong  light  is  re- 
flected from  the  polished  surface.  This  is  due  to  repeated 

u 


290  Systematic  Mineralogy.         [CHAI-.  XIII. 

parallel  twinning  producing  a  finely  lamellar  structure,  the 
laminae  of  which  act  in  the  same  way  as  the  lines  in  a  diffrac- 
tion plate.  The  biaxial  Mica  of  South  Burgess,  Canada, 
shows  a  similar  but  more  sharply  defined  star  when  held  in 
front  of  a  candle  flame  or  other  luminous  point.  This  is  at- 
tributed to  the  inclusion  of  minute  crystals  of  uniaxial  Mica 
arranged  along  lines  crossing  at  60°.  The  same  thing  may 
be  seen  in  thin  sections  of  Labradorite,  Aragonite,  Proustite, 
Brookite,  and  generally  in  transparent  minerals  that  either 
contain  minute  crystals  of  other  substances  symmetrically 
enclosed,  or  whose  crystals  show  repeated  parallel  twinning. 
Thin  plates  of  Aragonite  and  Strontianite,  when  held  at  a 
distance  of  about  six  or  eight  feet  from  a  candle  flame,  often 
show  an  extended  series  of  diffraction  spectra  on  either  side 
of  the  central  image. 

Fluorescence,  or  the  property  of  rendering  visible  rays  of 
higher  refrangibility  than  are  ordinarily  apparent  in  white 
light,  is  not  a  very  common  property  of  natural  minerals, 
but  it  is  well  marked  in  certain  varieties  of  Fluorspar,  es- 
pecially the  large  transparent  crystals  from  Weardale,  which 
transmit  an  emerald  or  grass-green  light,  but  reflect  a 
deep  sapphire-blue.  More  striking  examples  are  furnished 
by  the  liquid  hydrocarbons  known  as  Petroleum,  which  are 
colourless,  or  transmit  various  tints  of  yellow  to  brown,  but 
by  reflection  show  the  light  blue  rays  above  the  violet  of  the 
ordinary  spectrum. 

Phosphorescence,  or  the  power  of  emitting  light  in  a  dark 
place  is  characteristic  of  a  certain  small  number  of  minerals, 
and  may  be  variously  developed  by  exposure  to  sunlight, 
friction,  heating,  or  an  electric  discharge.  The  first  method 
is  completely  effective  with  Diamond  and  some  varieties  of 
Zincblende,  and  less  perfectly  so  with  Strontianite  and  other 
earthy  carbonates.  The  second  method  is  applicable  to 
Quartz,  two  pieces  of  rock  crystal  or  any  other  variety  of 
this  mineral  emitting  a  pale  yellow  light  when  rubbed  to- 
gether in  the  dark.  This  is  still  more  markedly  shown  by 


CHAP.  XIV.]  Phosphorescence.  291 

loaf  sugar.  Heating  is,  however,  the  most  generally  effective 
method,  as  many  minerals  which  are  rendered  phospho- 
rescent when  their  temperature  is  raised  are  not  so  affected 
by  sunlight  Fluorspar,  when  heated  to  200  or  300  degrees 
Cent.,  becomes  strongly  luminous,  the  light  being  usually 
blue  ;  but  with  the  green  variety  known  as  Chlorophane  it 
is  of  a  brilliant  emerald  green.  Phosphorite  in  the  same  way 
emits  a  yellow  light.  Topaz,  Diamond,  Calcite,  and  some 
silicates  also  phosphoresce  when  heated,  but  to  a  less  degree 
than  the  typical  examples,  Fluor  and  Phosphorite.  Phospho- 
rescent minerals  generally  lose  that  property  when  strongly 
heated,  but  it  may  be  more  or  less  restored  by  subjecting 
them  to  a  series  of  discharges  from  an  electrical  machine. 

Crookes  has  shown  that,  when  exposed  to  electric  cur- 
rents of  high  tension  in  an  extremely  rarified  atmosphere, 
Ruby  and  Sapphire  phosphoresce  with  intense  red  and  blue, 
and  Diamond  with  a  vivid  green  light. 

The  best  examples  of  phosphorescence  are  afforded, 
however,  by  the  sulphides  of  the  earthy  metals — Barium, 
Strontium,  and  Calcium — which,  though  not  natural  minerals, 
are  prepared  by  heating  the  native  sulphates  of  these  metals 
with  carbon.  Sulphide  of  Barium  is  the  so-called  Bologna 
phosphorus,  and  was  the  first  substance  in  which  the  pro- 
perty of  phosphorescence  by  sunlight  was  discovered.  These 
substances  are  applied  in  the  production  of  clock  faces, 
which  emit  sufficient  light  to  show  the  time  in  the  dark. 


CHAPTER  XIV. 

THERMAL  AND   ELECTRICAL   PROPERTIES   OF   MINERALS. 

Thermal  relations  of  minerals.  A  crystallised  substance 
placed  in  the  path  of  a  pencil  of  rays  emitted  from  a  heated 
body  affects  the  latter  in  the  same  way  as  it  would  a  beam 
of  light — that  is,  it  may  either  transmit  them  freely  without 


292  Systematic  Mineralogy.        [CHAP.  XIV. 

sensible  absorption,  or  it  may  absorb  them  entirely  or  in 
part  The  former  case,  of  which  Rock  Salt  is  the  most 
complete  example,  corresponds  to  heat  transparency,  or 
diathermancy ',  and  the  latter,  exemplified  by  Alum,  to  heat 
opacity,  or  athermancy.  Sulphur  and  Fluorspar  are  also 
diathermanous,  but  less  perfectly  so  than  Rock  Salt,  while 
Tourmaline,  Gypsum,  and  Amber  are  nearly  as  opaque  to 
heat  as  Alum.  Heat  rays  are  also  subject  to  the  same 
laws  of  reflection,  refraction,  and  polarisation,  as  those  of 
light,  being  refracted  singly  by  Rock  Salt  or  other  isotropic 
substances  of  sufficient  heat-transparency,  and  doubly  by 
those  crystallising  in  the  anisotropic  systems.  A  parallel 
beam  of  heat-rays  from  a  Rock  Salt  lens  falling  upon  two 
Mica  plates  is  more  completely  transmitted  by  the  latter 
when  their  planes  of  polarisation  are  parallel  than  in  any 
other  position,  the  amount  being  reduced  to  a  minimum 
when  they  are  crossed,  showing  an  extinction  exactly  analo- 
gous to  that  of  luminous  rays  under  similar  conditions. 

The  thermal  conductivity  of  amorphous  or  cubic  minerals 
is  in  like  manner  similar  in  any  direction,  while  in  those 
crystallising  in  the  other  systems  it  varies  with  the  crystallo- 
graphic  symmetry,  being  more  perfect  in  some  directions 
than  others,  or,  in  other  words,  such  minerals  show  axes  of 
conductivity  analogous  to  those  of  form  and  optic  elasticity. 
The  principal  investigations  upon  this  subject  are  those  of 
De  Senarmont,  who  determined  the  conductivity  in  sec- 
tions of  crystals  cut  in  known  directions  by  coating  one 
surface  with  wax,1  and  inserting  into  a  hollow  in  the  centre 
the  point  of  a  platinum  wire  heated  by  a  lamp  at  some 
distance.  When  the  heat  received  is  transmitted  equally 
in  all  directions,  the  wax  will  be  melted  in  a  circular  patch 
around  the  wire ;  but  if  the  rate  is  unequal,  it  will  be  an 
ellipse  whose  axes  will  correspond  to  those  of  maximum 

1  A  ratification  of  the  apparatus  in  which  the  heating  is  effected 
by  an  electric  current  is  shown  in  TyndalPs  Heat,  a  Mode  of  Motion* 
p.  202,  4th  edition. 


CHAP.  XIV.]  Dilatation  by  Pleat.  293 

and  minimum  rates  of  transmission  in  the  particular  section 
under  investigation.  In  Quartz  the  conductivity  is  higher 
in  the  direction  of  the  optic  axis  than  at  right  angles  to  it, 
while  in  Calcite  it  is  less,  the  two  minerals  in  regard  to  this 
property  being  positive  and  negative  in  the  same  way  as 
they  are  optically.  In  the  rhombic  system  the  three  crystal- 
lographic  axes  are  also  axes  of  dissimilar  thermal  conduc- 
tivity, while  in  the  oblique  and  triclinic  systems  the  latter, 
like  the  optic  axes,  are  not  directly  related  to  the  axes  of 
form. 

The  dilatation  of  crystallised  substances  by  heat  is  also, 
in  all  but  those  of  cubic  symmetry,  attended  by  change  of 
form,  as  the  rate  of  expansion  may  be  much  greater  in  one 
principal  crystallographic  direction  than  another.  This 
subject  has  been  elaborately  investigated  by  Fizeau,  by  a 
method  of  extraordinary  delicacy,  in  which  a  plane  surface 
of  the  crystal  under  investigation  is  covered  by  a  very  slightly 
concave  glass  plate,  the  curvature  being  so  large  as  to  allow 
the  formation  of  Newton's  rings  when  the  surface  is  illumi- 
nated by  a  monochromatic  yellow  light.  The  crystal  is 
supported  upon  a  tripod  of  platinum,  and  heated  in  an  air- 
bath,  when  if  the  expansion  be  uniform  the  surface  will 
approach  the  glass  regularly,  and  the  rings  change  their 
positions  symmetrically,  but  if  it  be  unequal  the  surface  will 
be  distorted,  and  the  distance  from  the  glass  will  change 
more  in  some  directions  than  others,  so  that  the  position  of 
the  rings  will  also  be  distorted,  and  by  careful  observation 
of  these  the  alteration  of  form  due  to  very  small  changes  of 
temperature  may  be  determined. 

The  general  conclusions  derived  from  these  experiments 
are  as  follows  : — 

Cubic  crystals  have  three  axes  of  dilatation  correspond- 
ing to  those  of  form,  the  coefficient  of  expansion  being 
similar  for  all,  and  consequently  for  any  direction,  in  the 
crystal. 

Uniaxial  crystals  have  a  principal  axis  of  dilatation 


294 


Systematic  Mineralogy'.         [CHAP.  XIV. 


corresponding  to  a  particular  coefficient  of  expansion,  which 
may  be  either  greater  or  less  than  that  in  directions  perpen- 
dicular to  it.  In  extreme  instances  the  expansion  may  be 
positive  in  one  direction  and  negative — that  is,  the  substance 
may  contract — at  right  angles  to  it ;  but,  in  any  case,  the 
arithmetical  mean  of  the  values  obtained  along  three  axes  at 
right  angles '  to  each  other,  will  correspond  to  that  observed 
along  a  line  inclined  at  54°  40'  to  them.  The  relation  of 
these  lines  is  that  of  the  trigonal  interaxes,  or  the  normals  to 
the  octahedral  faces  to  the  three  principal  axes  of  the  cube. 
In  the  rhombic  system,  the  axes  of  dilatation,  like  those 
of  optic  elasticity,  correspond  to  those  of  form ;  in  the  oblique 
system,  one  of  the  former  is  parallel  to  the  axis  of  symmetry, 
the  orthodiagonal,  but  the  others  make  different  angles,  not 
only  with  the  other  crystallographic  axes,  but  with  those  of 
optic  elasticity  and  thermal  conductivity.  The  relation  of 
these  different  directions  for  Orthoclase  is  shown  in  fig.  373, 
taken  from  Fizeau's  memoir,  which  re- 
presents the  distribution  of  two  of  each 
of  the  three  kinds  of  axes  ;  E2  and  E3 
being  two  of  the  axes  of  optic  elasticity, 
the  first  and  second  median  lines  of  the 
optic  axes  ;  D2  and  D3  two  of  the  three 
axes  of  dilatation,  c2  and  C3  two  of  the 
axes  of  conductivity,  and  A  A  the  clino- 
diagonal,  and  c  c  the  vertical  axis.  In 
Gypsum  the  direction  of  the  two  axes 
of  dilatation  and  the  optic  median  lines 
very  nearly  coincide.  In  the  triclinic 
system,  no  determinations  of  the  axes  of  dilatation  have  been 
made,  but  from  the  analogy  of  those  of  optic  elasticity  they 
probably  have  no  simple  relation  to  the  axes  of  form. 

As  a  consequence  of  their  unequal  linear  dilatation  in 

1  These  are  in  the  tetragonal  system  three  crystallographic  axes, 
and  in  the  hexagonal,  the  principal  axis,  one  lateral  axis,  and  the  inter- 
axis  at  right  angles  to  the  latter. 


CHAP.  XIV.]  Dilatation  by  Heat.  295 

different  directions,  the  geometrical  characters  of  anisotropic 
crystals  are  subject  to  change  with  alterations  of  temperature. 
For  example,  Calcite  elongates  in  the  direction  of  the  prin- 
cipal axis,  and  contracts  at  right  angles  to  it,  so  that  the 
angle  of  the  polar  edges  of  the  rhombohedron  becomes 
more  acute  by  heating.  The  amount  of  this  alteration  was 
found  by  Mitscherlich  to  be  8'  for  a  range  of  90°,  the 
observed  angle  at  10°  being  105°  4',  while  at  100°  it  was 
only  104°  56'.  Aragonite  and  Gypsum  are  examples  of 
minerals  in  other  systems  whose  rates  of  linear  expansion  in 
different  directions  are  sufficiently  different  to  show  varia- 
tions in  the  angles  when  heated.  In  the  greater  number  of 
instances,  however,  the  alterations  are  too  small  to  be 
directly  measurable,  but  the  principle  is  important,  as  es- 
tablishing the  difference  between  substances  crystallising  in 
different  systems,  even  when  they  may  have  forms  of,  exactly 
similar  geometrical  characters.  For  example,  the  coinci- 
dence between  a  cube  and  a  rhombohedron  of  90°  is  only 
true  for  one  particular  temperature,  as  the  latter  will  become 
either  more  acute  or  more  obtuse  when  heated  or  cooled, 
according  to  the  molecular  arrangement  of  its  substance, 
while  a  cube  is  rectangular  at  all  temperatures. 

The  parameters  of  the  unit  or  fundamental  form  of  a 
series  in  like  manner  are  altered  by  heating,  but  as  the  same 
change  extends  to  all  the  forms  of  the  series,  these  rela- 
tions are  not  changed  thereby.  For  instance,  if  the  para- 
meters of  the  form  (i  i  i)  be  changed  by  heat  from  a  :  b  :  c 
to  a'  :  b'  :  S,  those  of  the  form  (221)  will  change  at  the 
same  temperature  from  a  :  b  :  2  c  to  a'  .  b'  :  2  c1,  but  the 
symbols  will  not  be  altered,  as  the  new  length  of  the  vertical 
axis  in  the  second  will  be  twice  the  new  lengths  of  the  others. 
This  is  expressed  in  the  statement  that  the  principle  of  the 
rationality  of  the  axes  is  independent  of  temperature. 

The  effect  of  heating  substances  is  generally  to  increase 
their  volume  and  diminish  their  density,  and  this  is  accom- 
panied by  a  change  in  optical  properties,  more  particularly 


296  Systematic  Mineralogy.        [CHAP.  XIV. 

in  the  refractive  power,  which  is,  as  a  rule,  reduced,  and  in 
biaxial  crystals,  where  the  principal  indices  alter  unequally, 
the  change  often  affects  the  position  of  the  optic  axes. 
These  changes  are  most  apparent  in  the  earthy  and  alkaline 
sulphates.  In  Barytes,  Celestine,  and  Sulphate  of  Potassium, 
the  inclination  of  the  optic  axis  is  increased,  but  in  Felspar 
and  Gypsum  it  is  diminished,  by  heating.  In  the  latter  mineral 
the  alteration  is  very  considerable,  even  when  the  tempera- 
ture is  but  slightly  changed  ;  the  apparent  angle  of  the  optic 
axes  which  lie  in  the  plane  of  symmetry  is  about  90°  for  red 
light  at  the  ordinary  temperature  of  the  air,  but  at  116°  it 
is  o,  or  the  substance  is  apparently  uniaxial.  At  higher 
temperatures  the  axes  diverge  again,  but  in  a  plane  nearly 
at  right  angles  to  the  original  one.  On  cooling,  the  same 
changes  take  place  in  the  reverse  order.  In  some  varieties 
of  Felspar  the  change  in  angle  of  the  optic  axes  is  to  some 
extent  permanent — that  is,  it  does  not  return  exactly  to  the 
original  value  if  the  crystal  has  been  exposed  to  a  red  heat. 
Electrical  properties  of  minerals.  All  minerals  become 
electric  by  friction,  but  the  positive  or  negative  character  of 
the  electricity  developed  varies  according  to  circumstances. 
For  testing,  an  electroscope  is  used,  consisting  of  a  light 
metallic  needle  with  a  knob  at  either  end,  and  suspended 
in  the  centre  by  a  thread  of  raw  silk,  or  balanced  upon  a 
steel  point  by  an  agate  cap.  This,  when  rendered  positively 
or  negatively  electric  by  a  rod  of  glass  or  sealing-wax,  is 
either  attracted  or  repelled  by  an  excited  mineral,  according 
as  the  latter  is  charged  with  opposite  or  similar  electricity. 
If  the  mineral  under  trial  happens  to  be  a  conductor  of 
electricity,  it  will  be  necessary  to  insulate  it  in  order  to 
obtain  an  effect  upon  the  electroscope.  Several  minerals 
may  be  rendered  electric  by  pressure,  Calcite  possessing 
this  property  in  the  highest  degree ;  a  cleavage  fragment  of 
Iceland  Spar,  when  slightly  pressed  between  the  fingers,« 
becoming  positively  electric.  Topaz,  Aragonite,  Fluorspar, 
and  Quartz  are  similarly  affected,  but  in  a  less  degree. 


CHAP.  XIV.]  Magnetic  Properties.  297 

Minerals  that  become  electric  by  change  of  temperature 
are  said  to  be  thermo-  or  pyro-electric.  Axinite,  Tourma- 
line, Calamine,  Topaz,  and  Boracite  are  among  those  that 
show  this  property  best.  When  the  crystals  show  opposite 
electricities  at  different  points,  they  are  said  to  be  polar,  and 
the  points  at  which  the  changes  of  electricity  are  observed 
are  poles.  These  cannot,  however,  be  distinguished  as 
positive  and  negative,  as  both  kinds  of  electricity  are  de- 
veloped at  either  pole  alternately,  one  during  heating,  and 
the  opposite  one  in  cooling.  It  is  customary,  therefore,  to 
call  the  points  which  are  rendered  positively  electric  by 
heating  or  negative  by  cooling,  '  analogue  poles,'  and  those 
becoming  negative  by  heat  or  positive  by  cold,  '  antilogue 
poles.'  The  positions  of  these  vary  in  different  crystals. 
In  the  prominently  hemimorphic  species,  Tourmaline  and 
Electric  Calamine,  the  property  of  thermo-electricity  is  most 
characteristically  developed,  and  are  the  poles  at  opposite 
ends  of  the  principal  axis.  Boracite  has  eight  poles  corre- 
sponding to  the  solid  angles  of  the  cube,  while  in  Quartz  the 
electric  poles  axe  situated  at  the  ends  of  the  three  lateral 
axes. 

Magnetic  Properties  of  Minerals.  Nearly  all  minerals 
containing  iron  are  magnetic,  but  in  very  unequal  degrees. 
Practically,  only  native  Iron,  Magnetite,  and  Pyrrhotine,  or 
Magnetic  Pyrites,  are  sufficiently  magnetic  to  affect  a  com- 
pass needle  upon  a  pivot,  but  with  a  more  delicate  astatic 
instrument  many  other  minerals,  including  ferrous  sulphates 
and  silicates,  show  slight  traces  of  magnetism.  Native 
magnetic  oxide  of  iron,  or  Lodestone,  is  always  strongly 
magnetic,  and  is  often  found  in  masses  showing  distinct 
polarity,  but  natural  magnets  capable  of  supporting  con- 
siderable weights  are  rare.  Masses  of  Magnetite  of  a  regu- 
lar figure,  not  naturally  polar,  may,  however,  be  rendered  so 
by  touch,  in  the  same  way  as  a  steel  bar  is  magnetised.  The 
allied  minerals,  Chromic  Iron  and  Franklinite,  are  generally 
magnetic,  but  it  is  not  certainly  known  whether  this  is  a 


298  Systematic  Mineralogy.          [CHAP.  XV. 

special  property  or  caused  by  finely  interspersed  Magnetite. 
Spathic  Iron  Ore,  or  Ferrous  Carbonate,  becomes  strongly 
magnetic  when  heated  to  redness  from  the  formation  of 
magnetic  oxide,  as  do  most  of  the  Sulphides  of  Copper  and 
Iron  when  fused  in  an  oxidising  atmosphere. 

For  common  purposes  of  testing,  a  magnetic  needle  with 
a  brass  or  agate  centre,  to  be  used  upon  the  same  pivot  as 
the  electroscope,  will  generally  be  sufficient.  When  in  use, 
it  should  be  covered  with  a  glass,  to  protect  it  from  the 
disturbing  action  of  currents  of  air. 


CHAPTER  XV. 

CHEMICAL   PROPERTIES   OF    MINERALS. 

IN  the  determination  of  the  nature  of  a  mineral  a  knowledge 
of  the  kind  of  matter  constituting  it — that  is,  of  the  elemen- 
tary substances  it  may  contain,  and  the  relative  proportion 
of  such  substances  to  each  other  when  more  than  one  are 
present — is  of  primary  importance.  The  technical  methods 
whereby  such  knowledge  is  attained  are  known  as  qualita- 
tive and  quantitative  chemical  analysis,  and  it  will  be 
assumed  that  the  reader  is  familiar  with  the  details  of  such 
methods,  as  laid  down  in  the  treatise  on  chemical  analysis 
by  Professor  Thorpe  in  this  series,  or  in  some  other  standard 
work  on  the  subject. 

A  certain  knowledge  of  qualitative  analysis  is  of  great 
use  in  the  identification  of  minerals  of  obscure  habit,  and 
for  this  purpose  a  course  of  simple  testing  by  methods  not 
requiring  the  resources  of  a  complete  analytical  laboratory 
has  been  developed  by  mineralogical  chemists,  the  results 
obtained  by  such  tests  being  given  in  systematic  descrip- 
tions of  minerals  under  the  head  of  chemical  characteristics, 
or  some  equivalent  term. 


CHAP.  XV.]  Blowpipe  Apparatus.  299 

The  methods  employed  in  testing  are  divisible  into  two 
groups,  the  first  involving  the  application  of  heat  to  minerals, 
either  alone  or  in  the  presence  of  certain  reagents,  being 
known  as  analysis  by  the  dry  way,  and  the  second,  where 
liquid  solvents  or  reagents  are  used,  as  the  wet  way.  In  the 
former,  or  dry  way,  the  mineral  under  examination  is  sub- 
jected to  the  flame  of  a  lamp,  urged  by  a  blast  of  air,  which 
is  usually  produced  by  the  mouth  blowpipe,  although  the 
blast  of  a  bellows  or  the  flame  of  a  Bunsen  burner  may  also 
be  used.  The  latter  arrangements  are,  however,  only  to  be 
found  in  laboratories,  while  the  blowpipe  is  for  the  mine- 
ralogist essentially  a  portable  instrument,  and  may,  with  the 
necessary  apparatus  and  fluxes,  be  packed  into  a  case  of  small 
volume,  forming  a  portable  laboratory,  ready  for  use  with 
very  small  preparation,  and  as  such  is  invaluable  to  the 
travelling  mineralogist 

The  most  convenient  form  of  blowpipe  is  that  of  Gahn, 
as  modified  by  Plattner.  This  consists  of  a  brass  or  German 
silver  tube,  from  8  to  9  inches  long,  diminishing  in  diameter 
from  one-third  of  an  inch  above  to  one-sixth  or  one-seventh 
below.  A  trumpet-shaped  mouthpiece  is  fitted  to  the 
larger  end ;  the  smaller  one  fits  air-tight  in  a  cylindrical 
chamber  about  half  an  inch  in  diameter,  from  which  a  jet- 
pipe  about  an  inch  long  projects  at  right  angles ;  this  is  also* 
slightly  coned,  from  one-sixth  to  about  one-tenth  of  an  inch 
diameter,  and  is  terminated  by  a  conical  nozzle  of  platinum 
perforated  with  a  hole  about  ^  inch  diameter.  For  some 
purposes,  a  second  nozzle,  with  a  larger  aperture,  is  de- 
sirable. All  the  parts  are  ground  together  so  as  to  fit 
air-tight  without  screws.  The  trumpet- shaped  mouthpiece 
is  more  convenient  in  use  than  the  cylindrical  form,  from  the 
support  given  to  the  lips  when  the  blowing  is  carried  on  con- 
tinuously for  some  time.  Of  the  other  forms  of  blowpipe, 
that  known  as  Dr.  Black's  is  the  best,  when  made  of  an  appro- 
priate size,  the  great  fault  of  most  of  the  cheaper  instruments 
being  the  disproportionately  small  diameter  of  their  tubes. 


300  Systematic  Mineralogy.  [CHAP.  XV. 

The  flame  used  with  the  blowpipe  may  be  either  that  of 
a  candle,  a  lamp,  or  gas  jet,  the  best  being  that  of  an  oil  lamp 
with  a  flat  wick,  the  top  of  the  wick  being  cut  with  a  forward 
slope  of  about  20°.  When  in  use,  the  jet  of  the  blowpipe 
is  held  parallel  to  the  slope  of  the  wick,  and  a  current  of  air 
is  forced  into  the  flame  by  the  action  of  the  muscles  of  the 
cheeks,  breathing  being  kept  up  through  the  nostrils  in  a 
manner  which,  though  difficult  of  description,  is  easily 
learned  with  a  little  practice.  When  the  jet  is  placed  so 
that  the  air  enters  the  flame  at  the  higher  side  of  the  wick, 
a  large  proportion  of  the  unconsumed  gases  of  the  dark 
interior  part  is  carried  forward,  producing  a  pointed  flame 
of  a  certain  brilliancy,  which  is  of  a  neutral  or  non-oxidis- 
ing character,  and  is  called  the  reducing  flame.  If,  however, 
the  point  is  laid  above  the  middle  of  the  wick,  so  that  the 
air  is  brought  into  contact  with  the  dark  part  of  the  flame, 
total  combustion  of  the  gases  is  produced,  and  the  resulting 
flame  is  small,  slightly  luminous — having  the  characteristic 
blue  colour  of  burning  carbonic  oxide,  and  much  hotter 
than  the  reducing  flame.  This  is  called  the  oxidising  flame, 
as  any  substance  heated  in  it  is  subjected  to  the  full  effect 
of  the  unconsumed  oxygen  of  the  blast  or  of  the  adjacent 
atmosphere.  The  maximum  oxidising  effect  is  obtained 
immediately  before  the  point  of  the  flame.  The  power  of 
producing  a  clean  flame,  or  one  that  is  entirely  reducing  or 
oxidising,  is  one  of  the  first  essentials  to  success  in  blowpipe 
work,  and  should  therefore  be  practised  by  the  learner,  with 
the  tests  recommended  by  Plattner.  These  are  borax  beads, 
saturated  with  molybdic  acid  and  oxide  of  manganese  re- 
spectively ;  the  former  becomes  colourless  when  melted  for 
some  time  in  an  oxidising  flame,  but  turns  black  in  a  reduc- 
ing flame  ;  while  the  latter  is  of  a  dark  violet  colour  in  the 
oxidising  flame,  which  is  entirely  discharged  by  the  reducing 
flame.  To  obtain  these  effects  perfectly  by  the  use  of  the 
blowpipe  alone,  considerable  nicety  of  manipulation  is  re- 
quired. It  is  also  essential  that  the  metallic  oxides  used 


CHAP.  XV.]  Blowpipe  Apparatus.  301 

should  be  pure,  especially  that  of  manganese,  which  must 
be  free  from  iron,  otherwise  the  action  of  the  reducing 
flame  will  be  obscured  by  the  bead  taking  a  more  or  less 
green  tint. 

The  apparatus  required  (in  addition  to  the  blowpipe  and 
lamp;  is  as  follows  : — 

A  pair  of  forceps  with  platinum  points,  closing  by  a 
spring.  Pieces  of  platinum  wire  about  three  inches  long, 
bent  into  a  loop,  about  one-eighth  of  an  inch  diameter,  at 
one  end.  A  small  platinum  spoon,  and  a  piece  of  platinum 
foil.  A  jeweller's  hammer,  small  bright  steel  anvil,  and  an 
agate  mortar.  Pieces  of  glass  tube,  about  a  quarter  of  an 
inch  bore,  in  lengths  of  three  inches,  open  at  both  ends,  and 
a  few  shorter  pieces  closed  at  one  end.  A  few  watch  glasses  ; 
a  small  bar  magnet,  which  may  have  a  chisel  point  at  one  end. 
Charcoal  for  supports  :  the  best  are  pieces  of  straight  grained 
pine  charcoal,  about  three  inches  long,  and  three-quarters 
of  an  inch  to  one  inch  square  ;  these  are,  however,  with 
difficulty  obtainable,  and  should  therefore  be  used  carefully. 
For  most  purposes,  artificially  moulded  blocks,  formed  of 
charcoal  powder  cemented  with  starch  and  subsequently 
carbonised,  are  sufficient ;  they  are  made  in  various  sizes, 
and  may  be  purchased  of  dealers  in  chemical  apparatus. 
Hard-wood  charcoal,  made  from  brushwood,  is  to  be  avoided, 
as  it  usually  decrepitates  when  heated,  besides  leaving  a 
large  amount  of  ash.  The  use  of  a  plate  of  aluminium  as  a 
support  for  minerals  giving  coloured  sublimates  has  been 
recommended  by  Colonel  Ross. 

The  most  essential  fluxes,  or  reagents,  are  : — 

Borax,  calcined,  but  not  fused  into  a  glass  ;  phosphorus 
salt  (ammonic  sodic  phosphate) ;  dried  carbonate  of  soda, 
which  must  be  free  from  sulphate  ;  nitre ;  bisulphate  of 
potassium ;  nitrate  of  cobalt ;  fluorspar  ;  cupric  oxide  ;  and 
test  papers,  both  of  litmus  and  turmeric.  In  addition  to 
these,  a  few  liquid  reagents,  such  as  hydrochloric  and  nitric 
acids,  and  ammonia,  are  useful,  although  it  is  better  to  dis- 


302  Systematic  Mineralogy.  [CHAP.  XV. 

pense  with  them  as  much  as  possible,  as  they  cannot  be 
easily  carried  when  travelling. 

The  course  of  operations  followed  in  a  systematic  ex- 
amination of  minerals  by  the  dry  way  is  the  following  : — 

i.  Heating  in  tube  closed  at  one  end.     A  fragment  of  the 
mineral  (generally  called  the  assay]  is  placed  at  the  bottom 
of  a  tube  closed  at  one  end,  and  heated  first  over  the  flame 
of  a  lamp,  and  subsequently  in  the  blowpipe  flame.  Hydrated 
oxides  and  salts  by  this  means  give  off  water,  which  con- 
denses in  visible  drops  on  the  cooler  surface  of  the  tube. 
Nitrates  and  the  higher  oxides  of  manganese  give  off  oxygen, 
which  can  be  recognised  by  the  ignition  of  a  glowing  splin- 
ter of  wood.     A  few  minerals,  such  as  Sulphur,  Arsenic,  An- 
timony, Mercury,  and  their  oxides  and  sulphides,  sublime 
without  residue ;   but  are  redeposited  in  the  case  of  anti- 
mony and  arsenic  in  the  form  of  black  metallic  mirrors,  a 
short  distance  above  the  assay,  while  the  sulphides  of  the 
latter  element  are  recognisable  by  the  red  or  yellow  colours 
of  their  sublimates.     The  higher  sulphides,  such  as  iron 
pyrites,  give  a  sublimate  of  sulphur  ;  arsenical  iron  pyrites 
gives  both  sulphur  and  arsenic,  recognisable  by  the  red  and 
yellow  sublimates,  corresponding  to  the  minerals  Realgar 
and  Orpiment     These,  however,  are  products  of  the  de- 
composition, and  do  not  exist  as  such  in  the  mineral.     Ara- 
gonite,  Calcite,  Magnesite,  and  Dolomite  give  off  carbonic 
acid,  with  the  formation  of  caustic  lime  and  magnesia,  when 
strongly  heated,  while  Siderite  leaves  a  residue  of  magnetic 
oxide  of  iron.    The  carbonates  of  zinc,  copper,  and  lead  are 
also  decomposed,  producing  oxides  of  the  metals.      The 
sulphates  of  alumina  and  ferric  oxide  give  off  sulphuric  and 
sulphurous  acids,  the  acid  character  of  the  vapours  being 
recognised  by  a  slip  of  test  paper  inserted  in  the  mouth  of 
the  tube. 

2.  Heating  in  the  open  tube.  The  fragment  of  mineral 
under  examination  is  placed  about  half  an  inch  above  the 
lower  end  of  the  tube,  which  is  held  in  a  slightly  inclined 


CHAP.  XV.]  Fusibility.  303 

position,  and  the  blowpipe  flame  is  directed  upon  it,  so  that 
the  assay  is  heated  in  a  full  current  of  air,  when  sulphides 
give  off  sulphurous  acid,  which  is  easily  recognised  by  its 
characteristic  odour.  Arsenides  and  selenides  give  the 
peculiar  odours  characteristic  of  Arsenic  and  of  Selenium, 
and  sublimates  of  their  oxides,  which  deposit  at  a  greater 
or  less  distance  from  the  assay,  according  to  their  degrees 
of  volatility.  Antimony  gives  a  similar  sublimate  of 
antimonious  acid.  Many  sulphides  which  do  not  give  an 
indication  of  sulphur  in  the  closed  tube  are  decomposed 
with  the  formation  of  sulphurous  acid  in  the  open  tube. 

3.  Fusibility.  For  this  test  a  fine  splinter  of  the  mineral, 
held  in  the  platinum  forceps,  is  exposed  at  the  point  of 
maximum  heat  in  the  oxidising  flame.  This  supposes  it  to 
be  very  refractory ;  but  in  some  instances  exposure  to  the 
flame  without  blowing  is  sufficient  to  effect  fusion.  The 
range  of  the  melting  points  of  minerals  being  very  great, 
even  excluding  those  that  are  fluid  at  ordinary  tempera- 
tures— water,  mercury,  &c. — it  has  been  proposed  by  Von 
Kobell  to  express  fusibility  by  a  scale  of  typical  minerals, 
analogous  to  that  employed  in  describing  hardness.  This 
scale  comprises  the  six  following  numbers  ;  but,  owing  to 
the  indefinite  nature  of  the  gradations  between  the  different 
numbers,  it  is  not  much  used  : — 

(1)  Antimony- Glance.     Fuses  readily  in  the  flame  of  a 
candle. 

(2)  Stilbite.     Fuses  without  the  help  of  the  blowpipe. 

(3)  Almandine  Garnet.     A  large  or  thick  fragment  can 
be  melted  before  the  blowpipe. 

(4)  Actinolite  Hornblende  of  Zillerthal.     Thin  splinter 
melts  in  extreme  point  of  the  oxidising  flame. 

(5)  Felspar  (Adularia  of  S.  Gothard).     Similar  to  No.  4; 
but  fuses  with  greater  difficulty. 

(6)  Bronzite  of  Kupferberg.     Thin  splinters   can   only 
be  rounded  on  the  edges. 

When  the  edges  of  a  splinter  cannot  be  rounded   or 


304  Systematic  Mineralogy  [CHAP.  XV. 

softened  in  the  hottest  part  of  the  flame,  the  mineral  is  said 
to  be  infusible.  This  statement  is  of  course  only  true  in 
relation  to  the  means  employed  :  many  substances  that  are 
infusible  by  the  mouth  blowpipe  can  be  easily  melted  by 
the  oxyhydrogen  blowpipe  or  other  powerful  sources  of 
heat 

The  manner  of  fusion,  as  well  as  the  character  of  the 
fused  mass,  is  a  point  of  considerable  importance,  and  must 
be  carefully  observed,  some  minerals  fusing  quietly  to  a 
glass  or  enamel,  while  others  intumesce  or  swell  up,  from 
loss  of  water  or  other  volatile  components,  and  become 
scoriaceous  or  slaggy  masses;  others,  again,  give  a  mass 
which  crystallises  on  cooling. 

It  is  important  in  trying  fusibility  to  direct  the  flame 
upon  a  fine  point  of  the  fragment  under  examination,  so  as 
to  induce  fusion  as  quickly  as  possible,  otherwise  an  oxidis- 
ing action,  interfering  with  the  result,  may  be  set  up.  Thus 
the  double  sulphides  of  copper  and  iron  are  fusible,  but  if 
heated  for  some  time  in  air  below  their  melting  point,  they 
lose  sulphur,  and  give  a  residue  of  oxide  of  copper  and 
magnetic  oxide  of  iron,  which  is  practically  infusible.  In 
the  same  way,  silicates  containing  ferrous  oxide,  although 
fusible,  may  by  calcination  be  partly  resolved  into  silica  and 
ferric  oxide,  both  of  which  are  infusible. 

When  trying  the  fusibility  of  easily  reducible  metallic 
minerals,  care  must  be  taken  that  the  points  of  the  platinum 
forceps  in  contact  with  the  assay  are  not  strongly  heated, 
as  an  alloy  of  the  platinum  with  the  more  fusible  metal 
may  result.  It  is  safer  in  such  cases  to  try  the  fusibility 
upon  charcoal. 

Flame  colour  tests.  These  may  often  be  observed  simul- 
taneously with  the  trial  of  fusibility,  the  splinter  of  mineral 
used  for  the  latter  purpose  in  some  instances  giving  a  charac- 
teristic colour  to  the  flame  when  first  exposed  to  the  point  of 
the  oxidising  flame.  It  is  more  satisfactory,  however,  to  make 
a  special  trial.  Strontium,  Lithium,  and  Calcium  minerals 


CHAP.  XV.]  Flame  Reactions.  305 

give  a  red  colour,  which  is  best  seen  when  the  mineral, 
after  being  previously  heated  to  redness  in  the  reducing 
flame,  is  moistened  with  hydrochloric  acid,  and  exposed  in 
the  outer  blue  envelope  of  the  flame,  without  blowing.  This 
is  especially  the  case  with  the  sulphates  of  barium  and 
strontium,  which  may  be  partially  reduced  to  sulphides  in 
the  reducing  flame  upon  charcoal ;  the  sulphides  being 
converted  to  chlorides,  by  moistening  with  hydrochloric 
acid,  give  compounds  which  are  eminently  volatile  at  the 
high  temperature  of  the  blowpipe  flame.  Lithium  gives  a 
brilliant  crimson  ;  calcium  a  yellowish  red  ;  and  strontium 
a  purplish  red  tint.  When  the  two  latter  substances  are 
together,  the  colour  due  to  lime  is  seen  first,  and  then  that 
of  strontia.  When  the  red  flame  is  examined  by  a  dark 
blue  glass  screen,  the  light  due  to  lime  and  lithia  is  almost 
entirely  extinguished,  while  that  of  strontia  is  but  little 
altered.  Sodium  compounds,  even  in  very  minute  quantity, 
colour  the  flame  a  deep  yellow,  which  completely  effaces 
the  light  due  to  other  volatile  bodies ;  the  yellow  colour 
may,  however,  be  cut  off  by  the  blue  glass. 

Minerals  containing  barium  give  a  feeble  yellowish  green 
colour  to  the  flame  when  very  strongly  heated,  especially 
when  moistened  with  hydrochloric  acid.  Copper  gives  a 
bright  emerald  green,  except  in  the  state  of  chloride, 
when  the  flame  is  blue  with  a  purple  border.  Phosphates, 
when  moistened  with  sulphuric  acid  and  exposed  to  the 
outer  part  of  the  flame,  show  a  momentary  coloration  of  a 
pale,  bluish  green  ;  and  borates,  when  similarly  heated,  give 
a  brighter  green  colour. 

Potassium  gives  a  very  characteristic  reddish  violet  colour 
to  the  flame,  which  is  completely  hidden  by  even  a  very 
small  proportion  of  sodium.  If,  however,  the  yellow  light  of 
the  latter  is  cut  off  by  the  blue  glass,  which  allows  the  violet 
rays  to  pass  freely,  a  small  quantity  of  potash  may  be  de- 
tected even  in  the  presence  of  a  relatively  large  amount  of 
soda. 


306  Systematic  Mineralogy.          [Ciur.  XV. 

In  order  to  determine  the  presence  of  alkalies  in  silicates 
which  are  not  decomposable  by  acids,  they  should  be  pre- 
viously heated  with  fluoride  of  ammonium  in  the  platinum 
spoon  to  decompose  them,  when  the  bulk  of  the  silica  is 
volatilised,  and  the  residue  can  be  examined  in  the  flame 
with  the  blue  glass.  Chloride  of  calcium  may  be  used  for 
the  same  purpose,  as  the  lime  coloration  is  but  slightly 
transmitted  through  the  glass. 

The  alkaline  metals  may  be  determined  with  more  cer- 
tainty by  the  spectroscope,1  the  characteristic  bright  lines 
being,  for  their  oxides — 

Sodium,  a  bright  yellow  line. 

Calcium,  one  green  and  one  red  line. 

Lithium,  a  red  line,  which  is  at  a  greater  distance  from 
the  soda  (D  line)  than  that  of  lime.  This  is  cut  off  by  the 
blue  glass. 

Potassium,  a  dull  red  line  beyond  that  of  Lithia ;  this 
passes  through  blue  glass. 

Strontium,  an  orange  line  near  the  D  line,  several  red 
lines,  and  a  blue  line. 

Barium,  a  group  of  several  green  lines  close  to  each 
other. 

The  rare  alkaline  metals,  Cassium  and  Rubidium,  are 
also  readily  detected  by  their  spectra  ;  while  with  the  ordi- 
nary tests  they  may  be  confounded  with  Potassium  and 
Lithium.  For  the  detection  of  boracic  acid  in  silicates,  such 
as  Axinite  and  Tourmaline,  the  mineral  is  heated  on  a 
platinum  wire,  with  a  mixture  of  Fluorspar  and  Bisulphate 
of  Potash,  when  the  characteristic  green  line  is  produced. 

Heating  on  charcoal.  For  this  purpose  a  rectangular 
prism,  cut  from  a  piece  of  straight-grained  charcoal,  having 
the  rings  of  growth  perpendicular  to  the  ends  and  parallel  to 
the  length  on  two  faces,  is  preferable  to  a  moulded  charcoal 

1  Browning's  small  direct-vision  spectroscope  with  photographed 
micrometer,  mounted  on  an  upright  pillar,  is  a  convenient  form  of 
instrument  for  this  purpose. 


CHAP.  XV.]  Heating  on  Charcoal.  307 

block.  A  slight  depression  being  made  by  scraping  the 
surface  with  the  point  of  a  knife  near  one  end,  the  assay 
fragment  is  placed  in  it,  unless  the  mineral  decrepitates  by 
heat,  when  it  must  be  previously  powdered,  and  the  flame 
is  directed  downwards  upon  this  point,  the  charcoal  being 
held  with  a  slight  upward  inclination,  with  its  length  parallel 
to  the  direction  of  the  flame.  The  points  to  be  observed 
are  fusibility  or  infusibility,  the  production  of  phosphorous 
or  arsenical  odours  in  the  same  manner  as  in  the  open  tube, 
change  of  colour,  or  the  production  of  an  alkaline  mass. 
The  latter  reaction  is  characteristic  of  Calcite  and  Aragonite, 
which,  when  strongly  heated,  leave  an  infusible  residue  of 
caustic  lime,  giving  an  alkaline  reaction  with  moistened 
turmeric  paper.  Many  sulphides  and  other  compounds 
containing  iron  are  converted  into  infusible  masses,  which 
show  magnetic  properties.  The  principal  special  reaction 
upon  charcoal,  however,  is  the  production  of  coloured  in- 
crustations of  the  oxides  of  volatile  metals,  which  are  pro- 
duced from  their  combinations,  either  with  or  without  an 
actual  metallic  residue.  These  incrustations  cover  the  sur- 
face of  the  charcoal,  commencing  at  a  distance  of  an  inch  or 
less  from  the  assay,  according  to  the  volatility  of  the  metal. 
The  deposit  produced  by  Zinc  minerals  is  yellow  when  hot 
and  turns  white  in  cooling  ;  that  of  Cadmium  of  a  brownish 
yellow  usually  more  or  less  irised,  neither  giving  a  globule 
of  metal.  Antimony  and  Bismuth  compounds  yield  brittle 
metallic  globules,  the  former  with  a  thick  white  incrusta- 
tion edged  with  blue,  and  the  latter  one  of  lemon  yellow 
colour.  Lead  minerals  are  easily  reduced,  giving  a  malleable 
globule  of  metal,  and  a  yellow  incrustation  ;  but  when 
notably  argentiferous,  the  incrustation  is  more  or  less  crimson 
at  the  inner  edge.  Minerals  containing  Tin,  Copper,  or  Silver 
as  principal  constituents  are  reduced  to  the  metallic  state 
without  producing  an  incrustation  on  the  charcoal. 

Tests  with  soda  upon  charcoal.     The  whole  of  the  reac- 
tions depending  on  the  reduction  of  metallic  minerals  and  the 

X  2 


308  Systematic  Mineralogy.          [CHAP.  XV. 

formation  of  coloured  sublimates  on  charcoal  may  be  facili- 
tated by  adding  a  small  quantity  of  carbonate  of  soda  to 
the  substance  under  examination,  which  forms  a  slag  with 
the  infusible  constituents  ;  and  with  minerals  containing  Tin 
a  small  quantity  of  cyanide  of  potassium  should  be  used 
to  facilitate  the  reduction  and  to  protect  the  reduced  metal 
from  oxidation.  The  most  convenient  way  of  applying  this 
flux  is  to  mix  the  assay  in  a  finely  powdered  state  into  a 
paste  with  about  its  own  volume  of  dried  carbonate  of  soda 
upon  the  palm  of  the  hand  by  adding  a  few  drops  of  water. 
The  mixture  is  made  and  removed  by  a  small  iron  spatula 
or  the  blade  of  a  knife,  and  the  paste  is  spread  over  the 
charcoal,  care  being  taken  to  heat  it  up  gradually,  so  as  to 
dry  it  before  exposing  it  to  the  full  strength  of  the  flame. 
When  sufficiently  large,  the  metallic  globules  produced  are 
tested  as  to  their  brittle  or  malleable  character  by  the  ham- 
mer and  anvil ;  but  when,  as  generally  happens,  they  are 
small  and  interspersed  through  a  mass  of  slag,  the  assay  with 
the  adjacent  portion  of  the  charcoal  must  be  removed  and 
pulverised  in  the  agate  mortar.  The  charcoal  is  then  re- 
moved by  carefully  washing  the  contents  of  the  mortar  in 
a  gentle  current  of  water,  leaving  the  reduced  metal,  which, 
if  malleable,  will  be  found  in  flattened  scales  or  spangles. 

Other  special  uses  of  soda  are  in  the  decomposition 
of  infusible  silicates,  the  detection  of  Manganese,  and 
of  Sulphur  in  insoluble  sulphates,  or  the  lower  sulphides 
that  do  not  give  a  sulphur  sublimate  when  heated.  When 
Quartz  or  an  infusible  silicate,  powdered  and  mixed  with 
carbonate  of  soda,  is  heated  on  charcoal,  the  mixture  effer- 
vesces from  the  escape  of  carbonic  acid,  and  a  fused  alkaline 
silicate  is  formed,  which  is  soluble  in  water,  and  may  be 
decomposed  by  mineral  acids  with  the  production  of  silica 
in  the  gelatinous  or  soluble  form,  which  may  be  rendered 
insoluble  by  heating  to  redness.  This  is  the  ordinary 
method  adopted  for  rendering  these  minerals  soluble  for 
analysis  in  the  laboratory,  and  the  method  may  be  sometimes 


CHAP.  XV.]  Tests  on  Charcoal.  309 

adopted  with  advantage  on  the  small  scale  by  the  blowpipe. 
Minerals  containing  manganese,  when  fused  with  soda  in  a 
full  oxidising  flame,  give  a  green  enamel-like  mass  of  manga- 
nate  of  sodium.  This  test  may  be  made  upon  charcoal,  but 
generally  some  nitre  is  added  to  the  mixture,  and  the  fusion 
is  effected  on  platinum  foil.  The  test  for  sulphur  consists 
in  fusing  minerals  such  as  the  sulphates  of  barium,  stron- 
tium, or  calcium  with  soda  upon  charcoal  in  the  reducing 
flame  until  the  bulk  of  the  melted  flux  is  absorbed  by  the 
coal,  the  slaggy  matter  remaining,  and  then  the  adjacent 
portions  of  the  charcoal  are  cut  out  and  placed  upon  a 
bright  coin  or  plate  of  silver,  with  the  addition  of  'a  few 
drops  of  water,  in  which  the  alkaline  sulphide  formed  dis- 
solves, producing  a  dark  brown  or  black  stain  of  sulphide  of 
silver  upon  the  metal.  This  is  a  very  simple  and  delicate 
test ;  care  must,  however,  be  taken  that  the  soda  used  is  free 
from  sulphates,  which  point  must  be  previously  determined 
by  testing  it  upon  the  metal  alone. 

Fritting  tests  on  charcoal.  Some  infusible  metallic  oxides 
which  under  ordinary  circumstances  are  colourless,  become 
coloured  when  moistened  with  a  solution  of  nitrate  of  cobalt 
and  strongly  heated  upon  charcoal.  The  colours  are  blue 
with  alumina,  flesh  red  or  pink  with  magnesia,  and  green 
with  oxide  of  zinc.  The  first  and  last  of  these  are  in  fact 
the  ordinary  Cobalt  blues  and  greens  used  as  water  colours. 
It  is  essential  for  this  test  that  the  substance  tried  should  be 
infusible,  or  otherwise  a  cobalt  blue  glass  or  enamel  will  be 
formed. 

Tests  with  verifiable  fltixes.  The  salts  used  for  this 
purpose  are  Borax  (di-sodic  borate)  and  microcosmic  salt 
or  salt  of  phosphorus  (ammonio-sodic  phosphate),  which 
melt  into  a  clear  glass  at  a  red  heat.  The  active  agents  are 
in  either  case  the  combined  equivalent  of  boracic  and  phos- 
phoric acids  respectively,  borax  being  an  acid  salt,  and 
phosphorus  salt,  though  a  neutral  or  bibasic  phosphate, 
loses  its  ammonia  when  heated,  becoming  a  bibasic  sodic 


3io  Systematic  Mineralogy.          [CHAP.  XV. 

phosphate  with  one  atom  of  phosphoric  acid  free.  The 
acids  themselves  may  therefore  be,  and  are  sometimes,  used 
instead  of  their  sodium  salts,  but  they  are  inconvenient  from 
their  excessively  hygroscopic  properties. 

The  sodium  salts  of  boracic  and  phosphoric  acids  have, 
when  melted,  the  power  of  dissolving  up  nearly  all  metallic 
oxides,  producing  glasses  which  are  characteristically  coloured 
by  even  very  minute  quantities  of  such  oxides  as  are  of 
strong  colouring  power.  The  test  with  borax  is  one  of  the 
most  useful  in  the  whole  series  of  blowpipe  operations,  and 
is  performed  by  melting  in  the  loop  of  a  platinum  wire  a 
bead  of  borax,  which  should  be  perfectly  colourless  both  hot 
and  cold.  A  small  quantity  of  the  mineral  to  be  tried,  pre- 
ferably in  a  fine  powder,  is  then  added,  the  bead  is  remelted 
in  one  or  other  of  the  blowpipe  flames  until  the  substance  is 
completely  dissolved,  when  the  'bead  is  allowed  to  cool,  and 
the  colour  due  to  the  particular  oxide  and  flame  employed 
will  be  recognised.  The  following  are  the  most  characteristic 
reactions  obtained  in  this  way  with  compounds  of  different 
metals  and  borax  : — 

Iron  :  In  reducing  flame  dark  (bottle)  green ;  in  oxidi- 
sing flame,  yellow  while  the  bead  is  hot,  but  becomes  nearly 
colourless  when  cooled. 

Manganese  :  Colourless  in  reducing  flame,  deep  violet  or 
amethyst  in  oxidising  flame. 

Chromium  :  Grass  green  in  both  oxidising  and  reducing 
flame. 

Uranium  :  Green  in  reducing,  yellow  in  oxidising  flame. 

Cobalt :  Deep  blue  in  both  flames.  This  colouration  is 
produced  by  an  exceedingly  minute  quantity  of  oxide  of 
cobalt,  and  is  probably  the  most  delicate  of  all  the  tests  for 
this  metal. 

Nickel :  Reddish  to  brown  hot,  yellowish  to  dark  red 
cold  in  oxidising  flame,  which  is  rendered  blue  by  an  ad- 
dition of  nitre.  In  the  reducing  flame  the  colour  disappears, 
the  bead  becoming  grey  with  finely  divided  metallic  nickel. 


CHAP.  XV.}  Tests  with  Borax.  311 

These  reactions  refer  to  chemically  pure  preparations  in  mi- 
nerals ;  they  are  generally  obscured  by  the  presence  of  cobalt. 
Copper:  Green  while  hot,  turning  blue  on  cooling  in 
oxidising,  and  sealing-wax  red  in  reducing  flame.  There  is 
a  great  difference  between  the  colouring  power  of  the  two 
oxides  of  this  metal,  the  bead,  which  is  perfectly  transparent 
in  the  oxidising  flame,  being  coloured  with  cupric  oxide, 
becomes  opaque  from  the  formation  of  cuprous  oxide  or 
suboxide  of  copper  in  the  reducing  flame.  This  result  is 
most  easily  obtained  by  heating  the  green  bead  on  charcoal 
or  touching  it  when  melted  with  a  piece  of  tinfoil  so  as 
to  detach  a  minute  globule  of  tin,  which  has  an  ener- 
getic reductive  action  upon  the  cupric  oxide.  In  this  way 
a  small  trace  of  copper  may  be  recognised  in  the  presence 
of  iron,  in  spite  of  the  strong  colouring  power  of  the  latter 
metal  in  the  reducing  flame. 

The  colours  obtained  with  beads  of  salt  of  phosphorus 
are  generally  similar  to  those  of  borax,  but  there  are  some 
characteristic  differences,  especially  in  the  reducing  flame. 
Thus  Iron  gives  a  yellow  or  reddish  tint  instead  of  the  dark 
green  obtained  with  borax ;  Vanadium  gives  a  yellowish 
brown  in  the  oxidising  and  green  in  the  reducing  flame, 
both  being  green  with  borax ;  Uranium  gives  green  in  the 
oxidising  flame. 

It  is  often  of  more  importance  for  determinative  purposes 
to  know  the  colours  given  by  minerals  containing  more 
•than  one  element  of  strong  colouring  power  than  that  of 
the  components  taken  separately  in  a  pure  state,  as  in  such 
cases  very  characteristic  reactions  are  given  by  the  combi- 
nation. This  is  specially  the  case  with  the  iron  compounds 
of  certain  metallic  oxides  ;  thus,  Tungsten  in  the  form  of 
tungstic  acid,  gives,  with  salt  of  phosphorus,  a  colourless  or 
yellow  bead  in  the  oxidising,  and  a  blue  one,  which  is  green 
while  hot,  in  the  reducing  flame  ;  but  if  Iron  is  present,  as 
in  Wolfram  (natural  tungstate  of  iron),  the  latter  colour  is 
changed  to  a  brownish  red. 


312  Systematic  Mineralogy.          [CHAP.  xv. 

Another  example  is  afforded  by  titanic  acid,  which,  when 
pure  as  in  Rutile,  Anatase  and  Brookite,  gives,  with  salt  of 
phosphorus,  a  bead  colourless  or  cloudy  in  the  oxidising, 
and  red,  passing  into  violet  when  cold,  in  the  reducing 
flame.  The  latter  colour  is,  however,  changed  to  brownish 
red  when  iron  is  present,  as  in  titaniferous  iron  ores,  and 
the  violet  colour  can  only  be  brought  out  by  the  addition  of 
tin  or  zinc  to  the  bead. 

The  presence  of  titanic  acid  in  iron  ores  may  be 
rendered  apparent  by  a  method  devised  by  Gustav  Rose, 
which  is  of  great  interest.  If  the  dark-coloured  clear  bead 
when  completely  saturated  in  the  reducing  flame  be  trans- 
ferred to  the  oxidising  flame,  it  loses  colour,  but  becomes 
opaque  from  the  separation  of  titanic  acid  TiO2,  which  is 
much  less  soluble  in  melted  phosphate  of  soda  than  the 
lower  oxide  Ti2O3.  By  flattening  the  cloudy  bead  between 
the  forceps  while  still  hot  it  may  be  rendered  translucent, 
and  when  examined  by  the  microscope  is  seen  to  contain 
minute  crystals,  having  the  characteristic  square-based  form 
of  Anatase.  These  are  perhaps  more  readily  seen  if  a  drop 
of  water  is  added  to  the  bead  on  the  glass  slide,  when 
the  vitrified  sodium  phosphate  dissolves,  releasing  the 
crystals,  which  remain  suspended  in  water,  and  are  more 
visible. 

Quartz  and  many  silicates,  when  heated  in  sodic  phos- 
phate, do  not  dissolve,  'but  leave  an  opaque  mass  usually 
called  a  silicious  skeleton.  This,  according  to  Gustav  Rose, 
is  actually  crystallised  silica  of  the  variety  of  low  specific 
gravity  known  as  Tridymite,  a  mineral  found  only  in  rocks 
of  volcanic  origin.  The  subject  of  the  microscopic  character 
of  crystals  formed  by  other  oxides  has  recently  received 
considerable  attention  from  Sorby  and  other  investigators, 
whose  researches  should  be  consulted  by  the  student. 

Tests  by  the  wet  way.  The  use  of  the  methods  of  quali- 
tative analysis  by  the  wet  way,  in  the  determination  of 
minerals,  is  necessarily  restricted  to  those  that  can  be  applied 


CHAP,  xv.]  Tests  by  the  Wet  Way.  313 

without  requiring  any  great  amount  of  apparatus  or  labora- 
tory appliances,  and,  as  a  rule,  should  only  be  used  when 
the  blowpipe  tests  are  insufficient.  They  are  therefore  of 
most  value  in  the  examination  of  silicates  and  other  insoluble 
compounds  of  the  earthy  and  alkaline  metals  which  are  not 
readily  recognisable  by  the  dry  way. 

The  most  useful  reagents  are  hydrochloric,  nitric,  and 
sulphuric  acids,  ammonia,  sulphide  of  ammonium,  *  caustic 
potash  or  soda,  *  phosphate  of  soda,  *  nitrate  or  chloride  of 
barium,  *  oxalate  of  ammonia,  *  nitrate  of  silver,  *  molybdate 
of  ammonia.  These,  with  the  exception  of  those  marked  *, 
are  liquids,  and  must  be  kept  in  stoppered  bottles,  while 
the  others  may  be  kept  either  as  solutions  or  in  the  dry 
state,  the  former  being  most  convenient  when  they  are  often 
used,  but  when  they  are  only  required  occasionally  it  is 
better  to  make  the  test  solutions  when  wanted.  In  every 
case  distilled  water,  or  such  as  is  Tree  from  sulphates  and 
chlorides,  is  to  be  used. 

The  principal  pieces  of  apparatus,  in  addition  to  those 
already  mentioned  as  requisite  for  use  with  the  blowpipe, 
are  a  few  porcelain  capsules,  the  largest  about  -2  inches 
across,  test  tubes,  a  few  small  beakers  and  funnels,  paper 
filters  up  to  2\  inches  diameter,  or  sheets  of  filter  paper, 
a  wire  filter  stand,  and,  if  possible,  a  small  platinum 
crucible. 

The  course  of  examination  followed  should  be  similar  to 
that  adopted  in  systematic  analysis  with  substances  of  un- 
known composition,  and  in  all  cases  the  mineral  should  be 
finely  powdered,  the  test  for  solubility  in  water  being  first 
applied.  Among  the  minerals  readily  soluble  are  the  dif- 
ferent alkaline  Chlorides  and  Sulphates,  the  Alums,  the 
Sulphates  of  Magnesium,  Zinc,  Copper,  Iron,  Nickel,  &c.  ; 
less  so  are  Sulphate  of  Calcium  (Gypsum),  and  Arsenious 
acid.  Hydrochloric  acid  is  next  used,  first  without  and 
then  with  the  aid  of  heat,  and  subsequently  the  same  acid 
in  a  more  concentrated  form  when  necessary.  Minerals 


314  Systematic  Mineralogy.  [CHAP.  XV. 

may  be  completely,  partially,  or  not  at  all  soluble  by  this 
treatment.  The  solution  is  attended  with  effervescence  in 
the  case  of  carbonates  and  sulphides,  carbonic  acid  and 
sulphuretted  hydrogen  being  respectively  evolved.  The 
latter  may  be  recognised  by  its  action  upon  a  slip  of  test 
paper  made  with  acetate  of  lead,  which  is  blackened  when 
exposed  to  the  current  of  gas  at  the  mouth  of  the  test  tube 
or  beaker  used  for  making  the  solution.  The  different 
carbonates  belonging  to  the  Calcite-Aragonite  group  vary 
considerably  in  their  solubility  in  acids.  Thus,  calcic  car- 
bonate in  both  forms  effervesces  readily  with  very  weak 
hydrochloric  or  even  acetic  acid,  while  those  of  magnesium 
and  iron,  as  well  as  the  more  complex  varieties  containing 
two  or  more  of  these  metals,  are  but  slowly  acted  upon  even 
by  strong  acid  in  the  cold.  Compounds  containing  the 
higher  oxides  of  manganese  dissolve  in  warm  hydrochlo- 
ric acid  with  evolution  of  chlorine.  Silicates  belonging  to 
the  Zeolite  group  are  decomposed  with  a  -separation  of 
silica  in  the  gelatinous  form,  while  others,  such  as  Labra- 
dorite,  leave  a  pulverulent  or  granular  residue  of  silica,  the 
metallic  bases  passing  into  solution  as  chlorides.  In 
doubtful  cases  the  effect  of  the  acid  may  be  established  by 
filtering  the  insoluble  residue,  and  testing  the  clear  solution 
with  ammonia  and  phosphate  of  soda  ;  if  these  reagents 
produce  no  precipitates,  the  mineral  has  not  been  acted 
upon. 

Nitric  acid  is  used  for  the  decomposition  and  solution 
of  the  higher  sulphides  and  arsenides,  and  of  native  metals, 
most  of  which  are  attacked  with  the  evolution  of  red  fumes  of 
peroxide  of  nitrogen,  the  production  of  nitrates  of  the  metals, 
and,  in  the  case  of  sulphides  of  sulphates,  with  a  separation 
of  sulphur.  Nitrates  being,  however,  inconvenient  for  most 
analytical  determinations,  it  is  customary  to  convert  them 
into  chlorides  by  adding  hydrochloric  acid,  and  evaporating 
to  dryness  previously  to  commencing  the  systematic  exami- 
nation of  the  solution. 


CHAP.  XV.]  Examination  of  Silicates.  315 

Gold,  Platinum,  and  the  allied  metals,  being  insoluble 
in  either  nitric  or  hydrochloric  acid  alone,  are  brought  into 
the  soluble  form  as  chlorides  by  the  use  of  aqua  regia,  a 
mixture  of  nitric  and  hydrochloric  acids,  or  any  similar 
combination  producing  free  chlorine.  Sulphuric  acid  is 
principally  useful  in  decomposing  fluorides,  producing  sul- 
phates and  hydrofluoric  acid.  Experiments  of  this  kind 
must  be  performed  in  a  leaden  or  platinum  vessel,  the 
finely  powdered  minerals  being  heated  with  strong  sulphuric 
acid :  the  mouth  of  the  crucible  is  covered  with  a  glass  plate 
protected  by  an  etching  ground  of  wax,  upon  which  a  design 
is  marked  by  removing  the  wax  with  a  steel  point.  Wherever 
the  glass  is  laid  bare,  it  is  corroded  by  hydrofluoric  acid 
vapour,  reproducing  the  lines  drawn  upon  the  glass  plate. 
Liquid,  or  rather  a  strong  watery  solution  of  hydrofluoric 
acid,  is  also  an  exceedingly  convenient  reagent  for  the 
decomposition  of  silicates  where  it  is  desired  to  obtain  the 
metallic  bases,  especially  those  of  the  alkalies,  the  whole  of 
the  silica  being  removed  as  hydrofluosilicic  acid  gas  ;  but  it 
cannot  be  used  safely  except  in  laboratories  provided  with 
good  ventilation,  owing  to  the  corrosive  action  of  its  vapour 
on  all  glass  and  most  metallic  objects. 

The  usual  method  of  decomposing  silicates  insoluble  in 
acids — by  fusion  with  carbonate  of  sodium  in  a  platinum 
crucible  over  a  spirit  lamp  or  Bunsen  gas  burner,  as  adopted 
in  quantitative  analyses — may  be  imitated  on  a  small  scale 
before  the  blowpipe,  the  fusion  being  effected  upon  charcoal 
instead  of  on  platinum.  In  such  cases  a  small  quantity  of 
borax  should  be  added  to  the  alkaline  flux  to  prevent  the 
latter  being  absorbed  by  the  pores  of  the  charcoal,  and  to 
form  a  well-defined  bead  of  the  fused  mass  that  can  be 
removed  from  the  charcoal  without  loss.  The  bead  so 
obtained  is  dissolved  in  dilute  hydrochloric  acid,  the  solu- 
tion evaporated  to  dryness,  and  the  residue  moderately 
heated,  which  renders  the  silica  insoluble,  while  Lime, 
Magnesia,  Alumina,  and  other  bases  may  be  removed  by 


316  Systematic  Mineralogy.  [CHAP.  XV. 

digestion  with  dilute  hydrochloric  acid  or  soluble  chlorides. 
The  following  are  among  the  most  characteristic  reactions 
obtained  by  the  wet  way  with  the  more  abundant  metallic 
oxides  and  mineral  acids,  and,  as  tests'  that  can  be  used 
without  requiring  much  refinement  of  apparatus  or  manipu- 
lation, are  useful  to  the  mineralogist. 

Alumina.  This  base,  when  contained  in  the  hydro- 
chloric acid  solution  of  a  silicate  after  the  silica  has  been 
separated  by  evaporation  and  heating,  gives  a  white  gelati- 
nous precipitate  with  ammonia,  which  is  soluble  in  solution 
of  caustic  potash.  When  both  Iron  and  Alumina  are 
present,  as  is  the  case  with  many  silicates,  potash  must  be 
used  to  precipitate  the  former  as  ferric  hydrate,  and  after 
filtration  the  Alumina  is  thrown  down  by  adding  carbonate 
of  ammonia  to  the  solution,  the  excess  of  potash  having 
been  previously  neutralised  by  hydrochloric  acid  The 
precipitate,  when  dried  and  ignited,  is  tested  with  cobalt 
solution  in  the  manner  described  on  p.  309. 

Iron.  The  solution  of  any  iron  mineral,  when  brought 
to  the  condition  of  a  ferric  salt  by  heating  with  a  few  drops 
of  nitric  acid,  gives  a  dark  blue  precipitate  with  a  drop  of 
solution  of  yellow  prussiate  of  potash.  The  ammonia  pre- 
cipitate is  rusty  brown  with  ferric,  and  green  with  ferrous 
salts,  the  latter  passing  into  the  former  when  exposed  to 
the  air. 

Lime.  In  concentrated  solutions  a  white  precipitate  of 
sulphate  is  produced  by  sulphuric  acid,  which  is,  however, 
sensibly  soluble  in  water,  so  that  this  reaction  cannot  be 
used  with  weak  solutions  ;  in  such  cases  a  precipitate  may 
be  produced  by  the  addition  of  alcohol.  Oxalate  of  am- 
monia is  the  most  sensitive  test,  giving  a  white  precipitate 
in  very  dilute  solutions,  after  other  bases  separable  by 
ammonia  have  been  removed. 

Magnesia.  This  base  may  be  recognised  in  the  ammo- 
niacal  solution  remaining  after  the  removal  of  Alumina, 
Iron,  Lime,  and  other  bases  by  ammonia  and  oxalate  of 


CHAP.  XV.]     Tests  for  Magnesia  and  Baryta.  317 

ammonia,  by  adding  solution  of  phosphate  of  soda,  which 
produces  a  crystalline  precipitate  of  ammonio-magnesium 
phosphate.  The  formation  of  the  precipitate  is  facilitated 
by  well  stirring  the  solution  with  a  glass  rod,  and  allowing 
it  to  stand  for  some  hours. 

In  the  systematic  examination  of  minerals  containing 
Silica,  Alumina  and  Iron  Oxides,  Lime  and  Magnesia,  when 
the  tests  are  applied  in  the  order  given,  the  whole  number 
of  bases  may  be  verified  in  the  same  solution.  It  is  essen- 
tial when  Magnesia  is  present  to  have  a  considerable  quan- 
tity of  chloride  of  ammonia  in  the  liquid,  in  order  to  keep 
it  dissolved  until  the  other  bases  are  separated,  otherwise  it 
may  be  precipitated  as  hydrate  of  magnesia  by  ammonia. 

Baryta.  This  base  is  recognised  by  the  insolubility  of 
its  sulphate  in  water,  the  smallest  trace  of  it  in  a  solution 
producing,  with  a  drop  of  sulphuric  acid  or  of  solution  of 
a  soluble  sulphate,  a  cloudy  precipitate,  which,  however, 
may  require  some  considerable  time  to  form  when  the 
quantity  is  very  small.  Strontia  behaves  similarly,  but  the 
sulphate  being  more  soluble  takes  a  longer  time  to  precipi- 
tate. When  a  solution  containing  both  of  these  bases  in 
hydrochloric  acid  is  evaporated  to  dryness,  ignited,  and 
digested  with  alcohol,  chloride  of  strontium  dissolves,  leaving 
a  residue  of  chloride  of  barium,  and  the  properties  of  both 
can  be  established  by  their  flame  reactions. 

Most  of  the  so-called  heavy  metals  are  remarkable  for 
giving  dark-coloured  precipitates  when  treated  with  sulphu- 
retted hydrogen  or  an  alkaline  sulphide,  and  they  are  divisible 
into  groups  according  to  the  condition  of  the  solution 
necessary  to  ensure  precipitation.  Thus,  Gold,  Silver,  Lead, 
Copper,  Bismuth,  Arsenic,  and  Antimony  can  be  separated 
as  sulphides  from  an  acid  solution,  while  with  Iron,  Nickel, 
Cobalt,  Manganese,  and  Zinc,  the  solution  must  be  alkaline, 
as  the  sulphides  of  these  metals  are  decomposed  by  acids. 
Aluminium  and  Chromium  are  also  members  of  the  latter 
group,  but  the  precipitates  formed  with  sulphide  of  ammo- 


318  Systematic  Mineralogy.          [CHAP.  XV. 

nium  are  not  sulphides,  but  hydrates  of  alumina  and  chromic 
oxide.  Whenever  sulphuretted  hydrogen  is  used,  the  solu- 
tion of  the  metallic  bases  must  not  contain  much  free  nitric 
acid,  as  in  that  case  no  precipitation  of  sulphides  will  be 
effected  until  the  oxidising  action  of  the  acid  upon  the 
sulphuretted  hydrogen,  which  is  attended  with  a  separation 
of  sulphur,  is  exhausted. 

The  colours  of  the  precipitates  produced  by  sulphuretted 
hydrogen  with  the  more  abundant  metals  are  as  follows  : — 

Copper         .  .  Brownish  black. 

Lead    .        .  .  Bluish-black. 

Antimony    .  .  Orange  red. 

Arsenic        .  .  Bright  yellow. 

Aluminium  .  .  White  gelatinous  hydrate  of  alumina. 

Chromium    .  .  Green  hydrate  of  chromic  oxide. 

Zinc     .        .  .  White. 

Manganese  .  .  Flesh  colour. 

Nickel  1  T,,    , 

„          >       .        .     Black. 
Cobalt  J 

The  steps  to  be  taken  for  the  separation  of  the  different 
metals  existing  in  a  mixture  of  precipitated  sulphides  will  be 
found  in  the  treatise  on  chemical  analysis  previously  men- 
tioned, and  it  will  not  be  necessary  to  go  further  in  detail 
upon  this  point  here,  as  for  most  purposes  required  by  the 
mineralogist  the  metals  contained  in  such  mixtures  may  be 
more  readily  determined  by  the  dry  way  with  the  blow- 
pipe. 

Silver,  when  in  solution,  may  be  detected,  even  when  in 
very  minute  quantity,  by  means  of  hydrochloric  acid  or  any 
chloride,  which  produce  a  white  curdy  precipitate  of  chloride 
of  silver  soluble  in  ammonia,  and  becoming  grey  or  violet, 
and-  ultimately  black  when  exposed  to  sunlight.  The  chloride 
is  readily  reduced  to  the  metallic  state  when  mixed  with 
carbonate  of  soda  and  heated  on  charcoal. 

Lead.     Solutions  of  this  metal  in  nitric  acid  give  a  white 


CHAP.  XV.]       Tests  for  Silver,  Gold,  Copper,  &c.       319 

precipitate  of  sulphate  of  lead  with  sulphuric  acid,  and  a 
crystalline  deposit  of  chloride  of  lead  with  hydrochloric 
acid,  insoluble  in  the  cold,  but  dissolving  readily  in  boiling 
water.  The  presence  of  silver  in  lead  can  be  best  deter- 
mined by  the  method  of  cupellation  on  bone-ash,  a  special 
modification  of  the  process  for  use  with  the  blowpipe  having 
been  contrived  by  Plattner.  In  this  way  very  minute  silver 
beads  having  the  characteristic  lustre  and  colour  of  the 
metal  may  be  obtained,  but  it  is  a  useful  precaution  to  verify 
their  properties  by  dissolving  them  in  nitric  acid  and  testing 
them  with  salt.  If  the  silver  should  contain  gold,  it  will 
remain  as  a  black,  insoluble  speck  in  the  acid  solution. 

Gold  may  generally  be  best  recognised  by  the  dry  way, 
but  when  in  a  moderately  concentrated  solution,  it  is  pre- 
cipitated as  a  brown  metallic  powder  by  sulphurous  and 
oxalic  acid  or  ferrous  sulphate.  In  a  weak  solution,  chlo- 
ride of  tin  produces  a  purple  or  ruby-red  coloration,  or  when 
sufficiently  concentrated,  a  substance  known  as  purple  of 
Cassius,  whose  colour  is  due  to  finely-divided  metallic 
gold. 

Copper.  Solutions  containing  this  metal  as  chloride  or 
sulphate,  when  slightly  acid,  are  decomposed  by  metallic 
iron  with  a  deposition  of  copper.  Thus  a  knife-blade  is 
coppered  in  a  few  minutes  when  plunged  into  such  a  solu- 
tion. Ammonia  gives  a  deep  blue  colour  to  the  solution 
when  the  proportion  of  copper  is  very  small,  and  yellow 
prussiate  of  potash  is  a  still  more  delicate  test,  producing  a 
brown  precipitate,  or,  when  only  the  minutest  trace  of 
copper  is  present,  a  reddish-brown  tint  to  the  liquid. 

Sulphuric  add.  The  test  for  this  acid  is  the  inverse  form 
of  that  for  baryta,  a  solution  of  nitrate  or  chloride  of 
barium  producing  a  white  precipitate  of  sulphate  insoluble 
in  acids. 

Hydrochloric  acid  is  detected  by  means  of  nitrate  of 
silver  solution,  which  produces  chloride  of  silver,  as  de- 
scribed under  the  head  of  Silver. 


320  Systematic  Mineralogy.          [CHAP.  XV. 

Phosphoric  acid.  When  in  minute  quantity,  this  may  be 
best  detected  by  adding  molybdate  of  ammonia  to  the 
solution,  which  must  be  acidified  with  nitric  acid  and 
heated  for  a  short  time,  when  a  yellow  precipitate  contain- 
ing about  6  per  cent,  of  phosphoric  acid  is  produced,  and 
subsides  slowly.  When  the  proportion  is  larger,  it  is 
separated  as  ammonio-magnesian  phosphate,  in  the  same 
way  as  described  for  magnesia,  except  that  some  soluble 
magnesian  salt,  as  sulphate  or  chloride  of  magnesium,  is 
added  instead  of  phosphate  of  soda.  This  test  may  be 
performed  in  the  presence  of  ferric  oxide  and  alumina 
if  a  small  quantity  of  citric  acid  is  present,  as  these  oxides 
are  not  precipitated  by  ammonia  when  organic  matter  is 
present 

Quantitative  analysis.  Although  the  methods  of  quali- 
tative testing  are  sufficient  in  the  greater  number  of  instances 
for  the  identification  of  well-established  minerals,  there  is  a 
large  remainder  even  of  these  which  cannot  be  so  identified, 
a  knowledge  of  their  actual  composition  being  required  in 
addition ;  and  the  same  holds  good  with  those  of  doubtful 
or  unknown  constitution,  especially  when  the  latter  are  from 
new  localities.  For  those,  therefore,  who  may  desire  to 
extend  the  field  of  mineralogical  knowledge,  a  practical 
acquaintance  with  the  methods  of  quantitative  analysis  as 
applied  to  minerals,  or,  at  any  rate,  to  the  more  abundant 
and  simply  constituted  species,  is  essential,  but  the  larger 
number  of  students,  who  may  require  only  a  good  sight 
knowledge  of  known  minerals,  may  be  content  to  accept 
the  results  furnished  by  analytical  chemists  without  further 
inquiry.  In  this  case,  however,  a  knowledge  of  the  method 
of  calculating  the  results  of  analyses,  or  the  deduction  of 
formulae  from  percentage  quantities,  will  often  be  found  of 
great  use. 

Constitution  of  Minerals.  Among  minerals  are  included 
not  only  elementary  substances,  but  combinations  of  two  or 
more  elements  forming  the  classes  of  compounds  known  as 


CHAP.  XV.]  Chemical  Composition.  321 

oxides,  sulphides,  acids,  bases,  haloid  salts,  oxysalts,  sulpho- 
salts,  double  salts,  anhydrides,  and  hydrates. 

Such  combinations,  being  fixed,  have  a  definite  constitu- 
tion. Thus,  quartz  invariably  contains  7  parts  of  silicon,  and 
4  of  oxygen ;  calcite,  10  of  calcium,  3  of  carbon,  and  12  of 
oxygen  ;  fluorspar,  20  of  calcium,  and  19  of  fluorine  ;  and 
so  on.  The  proportion  of  the  different  components  being 
constant  in  those  forms  that  are  normally  constituted,  de- 
viations from  the  normal  types  can  be  shown  to  be  caused 
either  by  foreign  substances  existing  as  mechanical  im- 
purities, or  by  partial  substitution  of  one  or  more  of  the 
component  elements  by  others  of  analogous  properties, 
according  to  the  laws  of  isomorphism.  In  the  former  case, 
the  proportion  between  the  essential  constituents,  notwith- 
standing their  absolute  diminution  in  quantity,  is  unchanged ; 
while  in  the  latter,  the  change,  though  less  simple,  consists 
in  the  substitution  of  one  element  for  another  in  the  propor- 
tions of  their  chemical  equivalents.  Thus,  a  specimen  of 
calcite  containing  10  per  cent,  of  silica,  clay,  or  other  con- 
stituents insoluble  in  hydrochloric  acid,  cannot  be  con- 
sidered as  differing  essentially  from  the  normal  composition, 
if  the  remaining  90  per  cent,  is  so  constituted  as  to  make 
up  the  proportion  10  :  3  :  12,  or  9  :  27  :  io'8,  which  are  the 
ratios  of  calcium,  carbon,  and  oxygen  in  calcite  when  in  a 
pure  state. 

Variation  in  composition,  due  to  the  second  of  the  above 
causes,  or  the  partial  substitution  of  analogous  elements,  is 
commonly  observed  in  the  same  mineral,  as  almost  all 
specimens  of  calcite  show  a  deviation  from  the  typical 
composition  by  containing  small  quantities  of  the  elements 
magnesium,  manganese,  iron,  or  zinc.  These,  however,  are 
held  to  be  in  partial  substitution  of  the  normal  amount  of 
calcium,  and  the  analyses,  when  interpreted  according  to 
the  theory  of  equivalent  proportions,  are  found  to  be  in 
accordance  with  the  normal  constitution. 

This  theory  supposes  every  element  to  have  a  combining 


322  Systematic  Mineralogy.  [CHAP.  XV. 

value,  or  quantity  peculiar  to  itself,  and  that  its  compounds 
with  other  elements  are  formed  either  in  the  ratio  of  that 
quantity,  or  of  one  or  more  simple  multiples  of  it.  Thus, 
supposing  A  and  B  to  be  two  elements,  they  may  form 
compounds  :  2 A  +B,  A  +  £,  2A  +  $B,  A +  28,  &c. 

It  is  further  supposed  that  analogous  compounds  may 
be  made  with  another  element  C :  2A  +  C,  A+C,  2A  +  ^C, 
A  +  2C,  in  which  the  values  of  A  remain  unaltered ;  and  in 
like  manner,  other  elements,  D,  F,  &c.,  may  be  substituted 
for  A,  giving  a  series  of  compounds  with  B  and  C,  in  which 
the  latter  are  unchanged.  The  special  quantities  of  the 
elements  so  substituted,  or  the  combinations,  are  said  to  be 
equivalents ;  and  if  the  weight  of  any  one  be  known  or 
assumed,  the  others  may  be  referred  to  it,  whereby  a  series 
of  constant  numbers  expressing  the  combining  proportions  of 
the  different  elements  is  obtained.  These  are  called  chemi- 
cal equivalents. 

In  forming  such  a  series  it  is  necessary  to  fix  upon  some 
one  element  as  a  basis,  and  this  choice  is  necessarily  an 
arbitrary  one.  For  this  purpose  the  elements  oxygen  and 
hydrogen  have  been  chosen  :  the  former  as  being  the  most 
abundant  element  in  nature,  and  the  latter  as  being  the 
lightest  In  the  oxygen  series  of  equivalents,  introduced  by 
Berzelius,  oxygen  was  taken  at  100,  with  the  result  of  giving 
inconveniently  large  values  to  most  of  the  elements,  especially 
to  such  metals  as  silver,  gold,  antimony,  &c.  In  spite  of 
this  drawback  it  was  for  a  long  time  current  in  France  and 
Germany,  and  is  used  in  the  greater  number  of  works  upon 
mineral  chemistry  of  the  first  half  of  the  present  century,  a 
period  which  has  been  more  prolific  in  discovery  in  this 
branch  of  science  than  those  immediately  preceding  or 
following.  In  England  it  has  been  customary  to  use 
Dalton's  scale,  upon  the  basis  of  which  hydrogen  is  assumed 
as  unity,  being  the  lightest  of  all  the  elements,  from  which, 
assuming  its  combination  with  oxygen  in  water  to  be  in  the 
proportion  of  single  equivalents  by  weight,  the  equivalent  of 


CHAP.  XV.]  Atomic  Weight.  323 

the  latter  is  found  to  be  8,  that  of  water  HO,  or  protoxide 
of  hydrogen,  9,  and  so  on.  Latterly,  however,  a  modifica- 
tion of  Dalton's  original  hypothesis,  founded  upon  the 
weight  of  equal  volumes  of  the  elements  when  in  the 
gaseous  form,  has  come  into  general  use  among  chemists, 
and  the  older  schemes,  founded  upon  considerations  of  weight 
alone,  have  been  practically  abandoned.  This  is  founded 
upon  the  proposition  known  as  Avogadro's  law:  namely  that 
equal  volumes  of  all  gases  contain  an  equal  number  of 
molecules.  By  the  term  molecule  is  meant  a  quantity  of  an 
element,  or  compound  of  elements,  capable  of  independent 
existence,  but  so  small  as  to  be  incapable  of  further  division. 
The  molecule  of  a  compound  is  a  complex  of  still  smaller 
portions  of  the  component  elements,  which  are  known  as 
atoms,  an  atom  being  defined  as  the  smallest  combining 
proportion  of  an  element.  A  molecule  must  therefore 
contain  at  least  two  atoms  which,  in  the  case  of  an  element, 
are  both  of  the  same  kind,  but  in  that  of  a  compound  are 
of  dissimilar  kinds  (or  those  of  the  constituent  elements). 

The  atomic  weight  of  an  element  is  the  weight  of  a 
volume  of  its  vapour  expressed  in  terms  of  the  weight  of  a 
similar  volume  of  hydrogen  ;  the  unit  weight  adopted  being 
that  of  one  cubic  centimetre  of  hydrogen  under  the  normal 
pressure  of  760  mm.  of  mercury  at  o°  Centigrade.  The 
molecular  volume  of  an  element  or  compound  is  that 
corresponding  to  two  volumes  of  hydrogen  ;  the  molecular 
weight  of  an  element  is  therefore  usually  double  its  atomic 
weight. 

When  elements  or  compounds  can  be  obtained  in  the 
state  of  gases,  their  molecular  weights  may  be  determined 
by  direct  experiment,  otherwise  a  vapour  density  must  be 
assumed  upon  considerations  founded  upon  analogies  drawn 
from  known  compounds  supposed  to  be  similarly  constituted. 
Such  determinations  must  necessarily  be  doubtful. 

The  atomic  weight  of  an  element  is  determined  from  the 
analysis  of  some  one  of  its  best  defined  and  most  stable 

Y  2 


324 


Systematic  Mineralogy.          [CHAP.  XV, 


compounds,  the  molecular  constitution  of  the  latter  being 
assumed  The  researches  of  Dulong  and  Petit  have  shown 
that  the  atomic  weights  of  elements  are  to  each  other  in- 
versely as  their  specific  heats. 

The  following  table  contains  the  atomic  weights  of  the 
elements  as  far  as  they  have  been  accurately  determined. 
The  symbol  prefixed  to  each  is  held  to  signify  an  atomic 
unit,  when  used  in  combination,  in  the  construction  of 
molecular  formulae. 

TABLE  OF  THE  ATOMIC   WEIGHTS   OF   ELEMENTS. 


Name 

Symbol 

Class 

Atomic 
weight 

Aluminium 

Al. 

II.   IV.  VI. 

27-3 

Antimony  (Stibium) 

Sb. 

III.  V. 

122 

Arsenic  . 

As. 

I.   HI.  V. 

75 

Barium  . 

Ba. 

II.   IV. 

137 

Beryllium  (Glucinum) 

Be. 

II. 

9-33 

Bismuth  . 

Bi. 

V. 

208 

Boron     . 

B. 

III. 

II 

Bromine 

Br. 

I.  III.  V.  VII. 

80 

Cadmium 

Cd. 

II. 

112 

Calcium  . 

Ca. 

II.  IV. 

40 

Carbon  . 

C. 

II.  IV. 

12 

Cerium  . 

Ce. 

IV. 

92 

Coesium  . 

Cs. 

I. 

133 

Chlorine 

CL 

I.  III.  V.  VII. 

35'S 

Chromium 

Cr. 

II.  IV.  VI. 

52 

Cobalt    . 

Co. 

JI.  IV. 

59 

Copper  (Cuprum) 

Cu. 

II. 

63-4 

Didymium 

Di. 

II. 

96 

Erbium  . 

Er. 

II. 

II2'6 

Fluorine  . 

Fl. 

I. 

19 

Gallium  . 

Ga. 

VI. 

69-8 

Gold  (Aurum) 

Au. 

I.  III. 

196 

Hydrogen 

H. 

I. 

i 

Indium  . 

In. 

III. 

"37 

Iridium  . 

Ir. 

II.  IV.  VI. 

198 

Iron  (Ferrum) 

Fe. 

II.  IV.  VI. 

56 

Iodine    . 

I. 

I.  III.  V.  VII. 

127 

Lanthanum 

La. 

II. 

93 

Lead  (Plumbum) 

Pb. 

II.   IV. 

207 

Lithium  . 

Li. 

I. 

7 

Magnesium 

Mg. 

II. 

24 

CHAP.  XV.] 


Table  of  Elements. 


325 


Name 

Symbol 

Class 

Atomic 
weight 

Manganese 

Mn. 

II.  IV.  VI. 

55 

Mercury  (Hydrargyrum)  . 

Hg. 

II. 

2OO 

Molybdenum  . 

Mb. 

II.  IV.  VI. 

92 

Nickel    .... 

Ni. 

II.   IV. 

59 

Nitrogen 

N. 

I.  III.  V. 

H 

Niobium 

Nb. 

V. 

94 

Oxygen  . 

0. 

II. 

16 

Osmium. 

Os. 

II.  IV.  VI. 

198 

Palladium 

Pd. 

11.   IV. 

1  06 

Phosphorus 

P. 

I.  III.  V. 

3* 

Platinum 

Pt. 

II.   IV. 

198 

Potassium  (Kalium) 

K. 

I.   III.   V. 

39 

Rhodium 

Rd. 

II.  IV.  VI. 

104 

Rubidium 

Rb. 

I. 

I05'5 

Ruthenium 

Ru. 

II.  IV.  VI. 

104 

Sulphur  .... 

S. 

II.  IV.  VI. 

32 

Selenium 

Se. 

II.  IV.  VI. 

79 

Silver  (Argentum)   . 

Ag. 

I.  III. 

1  08 

Silicon  (Silicium)     . 

Si. 

IV. 

28 

Sodium  (Natrium)  . 

Na. 

I.  III. 

23 

Strontium 

Sr. 

II.  IV. 

88 

Tantalum 

Ta. 

V. 

182 

Tellurium 

Te. 

II.  IV.  VI. 

128 

Thallium 

Tl. 

I.  III. 

204 

Thorium 

Th. 

IV. 

234 

Tin  (Stannum) 

Sn. 

II.   IV. 

118 

Titanium 

Ti. 

11.  IV. 

248 

Tungsten                  .        . 
Wolfram          ... 

Tu.-1 
W.J 

IV.  VI. 

184 

Uranium 

u. 

II.  IV.  VI. 

240 

Vanadium 

V. 

III.   V. 

5i-4 

Yttrium  .... 

Y. 

II. 

617 

Zinc        .         .         .         . 

Zn. 

II. 

65 

Zirconium        .         . 

Zr. 

IV. 

90 

[Elements  marked  I.  are  monads  ;  II.  dyads  ;  in.  triads;  IV.  tetrads; 
v.  pentads;  VI.  hexads ;  and  vil.  heptads.] 


The  elements  are  classified  according  to  their  atomicities, 
or  combining  power,  as  measured  by  the  number  of  hydro- 
gen atoms  with  which  they  combine  to  form  definite  com- 


326  Systematic  Mineralogy.  [CHAP.  XV. 

pounds.  Thus  hydrogen  unites  in  the  following  manner, 
with — 

Chlorine,  i  equivalent  to  form  hydrochloric  acid  HC1. 

Oxygen,    2          „  „         water  HzCX 

Nitrogen,  3         „  „        ammonia  H3N. 

Carbon,    4         „  „         marsh  gas  H4C. 

— in  which  instances  the  quantities  i,  2,  3,  4  express  the 
combining  power,  or  quantivalence,  of  the  respective  ele- 
ments, which  are  said  to  be  monad  (i.),  dyad  (it.),  triad  (HI.), 
or  tetrad  (iv.),  according  to  the  special  number  of  hydrogen 
atoms  attached  to  them.  Besides  these,  there  are  higher 
ratios  of  quantivalence,  known  as  pentads  (v.),  hexads  (vi.), 
and  heptads  (vn.).  In  the  table  the  different  elements  are 
classified  by  these  numbers,  and  it  will  be  seen  that  one 
element  may  have  several  distinguishing  atomicities,  or 
that  it  may  form  several  different  types  of  compounds. 
Thus,  sulphur  is  dyad  in  sulphuretted  hydrogen,  H2S; 
tetrad  in  sulphurous  anhydride,  O2S  (oxygen  being  dyad) ; 
hexad  in  sulphuric  anhydride,  O3S. 

The  hexad  forms  of  chromium,  aluminium,  manganese, 
iron,  nickel,  and  cobalt  are  typified  in  the  compounds 
known  as  sesquioxides,  or  those  containing  two  equivalents 
of  metal  to  three  of  oxygen,  or  i  :  i^.  In  such  cases  the 
double  equivalent  of  the  metal  is  usually  considered  as  a 
unit,  and  is  represented  by  a  barred  symbol,  whose  atomic 
weight  is  double  that  of  the  ordinary  atom.  Thus  Al  repre- 
sents A12 ;  ¥e,  Fe2 ;  Al  O3,  A12  O3 ;  £e  C13,  Fe2  C13,  &c. 
This  arrangement  is  specially  convenient  in  representing  the 
composition  of  minerals  where  the  same  metal  occurs  in  two 
states  of  combination,  as  is  often  the  case. 

Construction  of  chemical  formula.  The  number  of  atoms 
of  the  different  elements  entering  into  the  constitution  of  a 
mineral,  when  its  composition  has  been  determined  by 
analysis,  is  found  by  dividing  the  percentage  proportion  of 
each  element  by  its  atomic  weight,  and,  subsequently, 


CHAP.  XV.]  Chemical  Formula.  327 

dividing  the  quotients  so  obtained  by  the  smallest  among 
them,  which  gives  a  series  of  numbers  standing  in  a  simple 
relation  to  each  other,  which,  when  reduced  to  whole  num- 
bers, give  the  number  of  atoms  required.  For  example, 
Baryte,  or  Heavy  Spar,  gives  the  following  analysis  : — 

Barium  58-80  .  .  137  .  .  ^^  =  0-429  =  i 
Sulphur  13-73  •  •  32  .  .  1J^P  =  0-428  =  i 
Oxygen  27-47  .  .  16  .  .  ^£-7  =  1-717  =:  4 

lOO'OO 

The  numbers  in  the  last  column  being  in  the  ratio 
i  :  i  :  4,  the  formula  required  is  BaSO4. 

When  three  or  more  elements  are  contained  in  a  mineral 
the  formula  obtained  by  writing  down  the  number  of  atoms 
of  the  constituents  side  by  side  is  known  as  an  elementary 
or  empirical  formula.  This  is  merely  the  simplest  nu- 
merical expression  obtainable,  not  expressing  any  opinion  on 
the  probable  arrangement  of  the  components.  In  most 
cases,  however,  it  is  necessary  to  obtain  some  idea  of  the 
grouping  of  the  constituents,  for  which  purpose  a  knowledge 
of  the  principles  of  chemical  classification  is  requisite  ;  and 
the  formulas  constructed  by  these  means  are  known  as 
rational  or  constitutional  formulae.  The  range  of  compounds 
occurring  in  minerals  is,  however,  comparatively  small.  It 
will  only  be  necessary  here  to  consider  the  principal  types 
of  composition  known  as  acids,  bases  and  salts.  According 
to  modern  views,  an  acid  is  a  compound  of  hydrogen  with 
an  electronegative  element  or  combination  of  elements 
known  as  a  compound  radical. 

Hydrochloric  acid  HC1,  hydrobromic  acid  HBr,  and 
hydrofluoric  acid  HF1  are  examples  of  the  first  kind,  or 
acids  with  simple  radicals.  The  acids  of  compound  radicals 
may  contain  either  oxygen  or  sulphur  in  the  radical,  but  they 
are  otherwise  analogous  in  constitution ;  the  former  are 
called  oxygen  acids,  and  the  latter  sulphur  acids.  They 


328  Systematic  Mineralogy.          [CHAP.  XV. 

are  further  distinguished  according  to  the  number  of  atoms 
of  hydrogen,  as  monohydric  with  one,  dihydric  with  two, 
or  trihydric  with  three  atoms. 

The  constitutional  formulas  of  acids  are  constructed  on 
the  hypothesis  of  containing  one,  two,  or  three  atoms  of 
oxygen  or  sulphur,  one  half  of  which  is  united  with  an 
equal  number  of  atoms  of  hydrogen,  and  the  other  half  with 
an  acid  radical,  which  may  be  either  mono-  di-  or  tri-valent, 
according  to  the  character  of  the  third  element.. 

Thus,  nitric  acid  has  the.  following  formula  : — 

Elementary.  Constitutional. 

HNO3  H-O-(NO2)' 

Sulphuric  acid   .         .     H2S04  H2=O2=(SO2)" 

Phosphoric  acid         .     H3PO4  H3=O3=(PO)'" 

Sulpho-carbonic  acid  .     H2CS3  H2=S=(CS)" 

where  the  accents  represent  the  equivalency  of  the  radical. 

When  the  molecule  of  an  acid  is  broken  up  by  the  re- 
moval of  the  whole  amount  of  hydrogen,  and  the  corre- 
sponding quantity  of  oxygen  required  to  form  water,  H2O, 
or  one  atom  of  the  latter  to  two  of  the  former,  or  in  the  case 
of  a  sulphur  acid  of  sulphur  to  form  sulphuretted  hydrogen, 
H2S,  a  compound  is  obtained  known  as  the  anhydride  of 
the  acid,  which  is,  in  fact,  an  oxide  of  the  radical.  In  the 
case  of  mono-  and  tri-hydric  acids  two  molecules  are  re- 
quired in  order  to  express  the  results  in  whole  numbers  of 
atoms.  Thus  : — 

Nitric  acid — 

(z  mol.)  or  2  (HNO3)  less  I  eq.  H2O  giving  (^Jo')  °  =  N2°5 

Sulphuric  acid — 

(I  mol.)      H2SO4    „     i        H2O    ,,        (SO2)  O   =  SOS 

Phosphoric  acid — 

(2  mol.)  2  (H3P04)    „     3        H20    „      (p£)  O3  =  P2O5 

Carbonic  acid — 

(i  mol.)    H2CO3      „     I        H2O      „        (CO)  O  =  CO2 

Sulpho-carbonic  acid — 

(i  mol.)     H2CS3       „     i         H2S      „          (CS)  S  =  CS2 


CHAP.  XV.]  Acids  and  Bases.  329 

The  compounds  in  the  last  column  are  therefore  called 
anhydrides  of  the  corresponding  acids,  or  generally  but  im- 
properly, anhydrides,  without  further  qualification.  In  the 
older  works  on  chemistry  they  are  called  anhydrous  acids, 
the  acids  defined  as  above  being  considered  as  hydrated 
acids. 

A  base  is  defined  to  be  a  combination  of  hydrogen  with 
a  compound  radical  of  an  electro-positive  character,  con- 
sisting of  an  electro-positive  element  or  metal  united  with 
oxygen  or  sulphur,  the  former  being  called  an  oxygen,  and 
the  latter  a  sulphur,  base. 

The  constitution  of  bases  is  represented  similarly  to 
that  of  acids,  the  constituent  atoms  of  oxygen  being  con- 
sidered as  combined  to  the  extent  of  one-half  with  an  equal 
number  of  hydrogen  atoms,  and  the  other  half  with  an 
equivalent  atom  of  the  metal,  producing,  as  in  the  case  of 
the  acids,  monohydric  and  polyhydric  bases,  or  hydroxides, 
or  with  sulphur  as  hydrosulphides,  thus  :  — 

i  i 

H  —  O  —  K      is     Hydropotassic  oxide,  or  of  the  type  (HO)  R 


n 


2  =  O2  =  Ba,,    Hydrobaric  oxide  ,,  (HO)2  R 


in 


H3  =  O3  =  Bi,,    Hydrobismuthic  oxide         ,,  (HO)3  R 

The  Roman  figures  represent  the  equivalency  of  the 
metallic  element,  and  in  the  type  formulae  in  the  last  column 
R  stands  for  any  metal  of  corresponding  equivalency. 

And  of  the  analogous  sulphur  compounds  :  — 

i  i 

H  —  S  —  K      is     Hydropotassic  sulphide,  or  of  the  type  (HS)   R 

H  II 

H2=S2  =  Ba,,     Hydrobaric  sulphide  ,,  (HS)2  R 

in  ni 

H3  =  S3  =  Bi  „     Hydrobismuthic  sulphide        ,,  (HS)3  R 

Basic  anhydrides  are  produced  in  the  same  manner  as 
the  acids  by  the  removal  of  the  hydrogen,  with  sufficient 
oxygen  or  sulphur  to  form  water  or  sulphuretted  hydrogen 


330  Systematic  Mineralogy.  [CHAP.  XV. 

from  the  hydrometallic  oxides  or  sulphides  respectively. 
Thus  :— 

2  HOK    —    H2O   =  K2O  known  as  potassic  oxide  or  potash. 
H2O2Ba  —    H2O   =  BaO        ,,         baric  oxide  or  baryta. 

»  -a  f\  TJ;      i  w  r»      TJ;  r»  /bismuth  oxide,  or  strictly  di- 

2H303Bi-3H20      Bi203        „         |     bismuthic  trioxide. 

And  from  the  analogous  sulphur  compounds—' 

2  HSK      —    H2S  =  K2S  or  dipotassic  sulphide. 

H2S2Ba  —    H2S  =  BaS   ,,  baric  sulphide. 
2  H3S3Bi  —  3H2S  =  Bi2S3  ,,  dibismuthic  trisulphide. 

The  last  of  these  is  the  mineral  known  as  Bismuth 
Glance.  The  anhydrides  of  bases  are  therefore  the  oxides 
and  sulphides  of  their  respective  metals. 

A  salt  is  considered  to  be  a  combination  formed  by  the 
action  of  equivalent  quantities  of  an  acid  and  a  base  upon 
each  other  when  the  whole  of  the  hydrogen  and  one  of  the 
atoms  of  oxygen  are  removed,  as  water.  Thus  : — 

Nitric  acid          .         .  H-O-(NO2)|       (K-O-fNO,,)  or  K  NOj 

>  =  J      potassic  nitrate,  and 
Hydropotassic  oxide  .  H  —  O  —  K        J       ( H2O  water. 

Sulphuric  acid   .         .  H2  =  O2  =  (SO-j)  ]       (Ba  =  O2=(SO2)  or  BaSO* 

\  =  \      baric  sulphate,  and 
Hydrobaric  oxide       .  H2  =  O2  =  Ba      j       (2  H2O  water 

Acids  and  bases  containing  like  amounts  of  hydrogen 
combine  in  equal  molecules,  but  when  they  are  unlike  the 
molecular  relation  of  acid  to  base  in  a  salt  is  dissimilar. 

Salts  formed  by  acids  of  a  simple  radical,  such  as  hydro- 
chloric acid,  HC1,  and  hydrofluoric  acid,  HF1,  are  called  Ha- 
loid Salts,  and  those  with  acids  of  compound  radicals  Oxy-salts 
and  Sulpho-salts.  A  neutral  salt  is  that  resulting  from  equi- 
valent quantities  of  acid  and  base  ;  it  is  also  called  a  normal 
salt.  An  acid  salt  is  a  combination  of  a  molecule  of  normal 
salt  with  one  or  more  molecules  of  acid ;  while  a  basic  salt 
is  similarly  a  normal  salt  combined  with  one  or  more  mole- 

f  K  SO   1 

cules   of  base.     Bisulphate  of  potassium   <  pr2cr)4  /•  is  an 


CHAP.  XV.]  Constitution  of  S'alts.  331 

example  of  the  former,  and  Malachite  or  basic  carbonate  of 
Copper  I  HUCuQ     I-  of  the  latter  class. 

Acid  and  basic  salts  may  in  some  cases  be  free  from 
hydrogen,  that  is,  they  may  consist  of  a  normal  salt  com- 
bined with  the  anhydrides  of  the  acid  or  base  respectively. 

Of  this  character  are  acid  bisulphate  of  Potassium  K2§94  "I 

SO3  / 

which  is  obtained  by  heating  the  salt  ^aSO*  j  ^  basic  chfo_ 
mate  of  Lead  2  pbQ  4  j  ,   and  the  oxychlorides  of  Lead 

PbO  2  }   and  2PbO  J   formmg  the  rare  minerals  Matlockite 
and  Mendipite. 

Double  salts  are  compounds  of  two  different  salts,  which 
may  be  either  similar  or  dissimilar  in  class  or  constitution. 


f 

Thus,  Carnallite  <  MgQ    is  a  compound  of  two  similar 

haloid  salts,  the  chlorides  of  Potassium  and  Magnesium  : 
Cryolite    <  ^  p,  .  of  the  two  haloids  of  dissimilar  consti- 


f  Na2SO 
tution;  Blodite  <  jy^gQ     of  two  similar  sulphates  ;  Potash 

K.  SO   1 
Alum  A1  CV.  4  >  of  two  dissimilar  oxysalts  (sulphates)  and 

Ala^U^  J 

Chlorapatite    r,a  T,2^    >   of  an  oxysalt  and  a  haloid,  phos- 
3^a3r  2w8  j 

phate  and  chloride  of  calcium.     The  acid  sulphate  of  po- 

K  SO   T 
tassium  u^cr^  >  might  also  be  regarded  as  a  double  sul- 

H2^V^4  J 

phate  of  potassium    and  hydrogen,  except  for  the  special 
signification  attached  to  the  term  acid. 

Many  minerals,  especially  alkaline  sulphates  and  other 
easily  soluble  salts,  give  off  water  with  more  or  less  readiness 
when  the  crystals  are  exposed  to  the  air,  in  some  cases 
without,  but  more  readily  by,  heat.  Such  water  is  usually 
regarded  as  not  essential  to  the  constitution,  or  as  water  of 


332  Systematic  Mineralogy.          [CHAP.  XV. 

crystallisation,  when  it  is  given  off  at  the  boiling  point  of 
water  or  a  little  above  it,  and  when  the  salt  so  dehydrated 
takes  the  same  amount  again  when  dissolved  and  recrystal- 
lised.  The  water  so  combined  is  often  distinguished  by 
the  symbol  Aq.  Thus,  Potash  Alum,  containing  24  equiva- 

K  SO    ") 

lents  of  water,  is  represented  by   Ai  c  ri4   >+24Aq:but 

-tt.l203Ui2  J 

when.a  high  temperature  is  requisite  to  drive  off  the  water, 
the  latter  is  to  be  regarded  as  the  result  of  decomposi- 
tion of  an  actual  hydrogen  compound  essential  to  the  con- 
stitution of  the  mineral.  For  example,  the  silicate  known  as 
Dioptase,  which  yields  by  analysis  oxide  of  copper,  water, 
and  silica  in  equivalent  proportions,  is  not  decomposed 
below  a  red  heat,  so  that  it  should  be  regarded  as  consisting 
of  H2CuSiO4  rather  than  as  a  hydrated  silicate  of  the  form 
CuSiO3+Aq.  Such  cases,  however,  are  often  of  doubtful 
interpretation,  depending  upon  the  consideration  of  com- 
pounds of  analogous  constitution  or  other  more  or  less 
assumed  data,  so  that  in  most  cases  it  is  simpler  to  accept 
the  result  of  analysis,  which  at  any  rate  expresses  the  fact 
that  water  has  been  found. 

The  class  of  minerals  known  as  hydrated  oxides  are 
similarly  of  uncertain  constitution,  as  they  may  be  regarded 
either  as  consisting  of  anhydrous  oxides  combined  with  one 
or  more  molecules  of  water,  or  as  compounds  of  bases 
(hydroxides)  with  their  anhydrides,  and  as  such  forming  a 
class  of  compounds  intermediate  between  acids  and  bases. 
For  most  mineralogical  purposes,  however,  the  former  is  the 
more  convenient  view. 


CHAP.  XVI.]  Heteromorphism.  333, 


CHAPTER  XVI. 

RELATION   OF   FORM  TO  CHEMICAL  CONSTITUTION. 

Heteromorphism. — When  a  mineral  is  of  well-defined  che- 
mical constitution,  its  physical  and  crystallographical  cha- 
racters are,  as  a  rule,  similarly  well  defined  ;  while,  on  the 
other  hand,  substances  that  are  only  known  in  the  amor- 
phous condition  are  very  generally  variable  in  composition. 
In  addition  to  these,  many  instances  are  known  where  a 
substance  of  particular  chemical  constitution  appears  in 
forms  belonging  to  different  crystallographic  systems,  and 
also  with  differences  in  physical  character.  This  property  is 
called  heteromorphism,  and  the  substances  possessing  it  are 
said  to  be  di-  or  tri-morphic,  according  as  they  appear  in 
two  or  three  different  crystalline  systems,  the  latter  being 
the  largest  number  yet  observed  in  any  heteromorphic 
mineral. 

These  different  varieties,  when  found  in  nature,  are 
spoken  of  as  different  minerals ;  but  when  they  are  arti- 
ficially produced,  no  such  distinction  is  made. 

The  following  are  among  the  more  remarkable  cases  of 
natural  heteromorphous  substances : — 

Sulphur. — This  element  is  known  in  three  conditions — 
i.  As  an  amorphous  plastic  substance,  produced  when 
melted  sulphur  heated  to  200°  Cent,  is  poured  into  water, 
having  the  sp.  gr.  1-92,  and  insoluble  in  bisulphide  of 
carbon  ;  2.  in  crystals  belonging  to  the  rhombic  system, 
combinations  of  rhombic  pyramids,  of  sp.  gr.  2*06,  which  are 
deposited  by  spontaneous  evaporation  from  a  solution  of 
sulphur  in  bisulphide  of  carbon  ;  and  3.  in  prismatic  com- 
binations belonging  to  the  oblique  system,  of  sp.  gr.  1-97, 
which  are  the  common  forms  of  sulphur  crystallised  from 
fusion.  Both  of  the  crystalline  varieties  are  of  the  charac- 
teristic yellow  colour  of  sulphur,  while  the  amorphous  one  is 


334  Systematic  Mineralogy  [CHAP.  XVI. 

dark  brown.  The  second  or  rhombic  form  is  the  only  one 
found  as  a  natural  mineral 

Carbon. — Also  in  three  conditions — i.  amorphous  in 
coal  as  the  deposit  produced  from  the  imperfect  combustion 
of  a  volatile  carbon  compound,  as  soot,  lamp  black,  &c. ; 
2.  crystallised  in  the  cubical  system  in  Diamond,  a  non- 
conductor of  electricity,  having  the  sp.  gr.  3-55  and  hardness 
10 ;  and  3.  crystallised  in  the  hexagonal  system,  as  Gra- 
phite, having  the  sp.  gr.  2-3,  and  hardness  1-5,  and  a  con- 
siderable degree  of  electric  conductivity. 

The  allied  elements,  Boron  and  Silicon,  have  been  shown 
by  Deville  to  occur  in  dimorphous  modifications  analogous 
to  those  of  Diamond  and  Graphite  in  carbon,  which  are 
distinguished  as  the  adamantine  and  graphitoidal  forms  of 
the  elements  respectively.  They  do  not,  however,  occur 
as  natural  minerals. 

Antimonic  oxide — Sb2O3,  is  both  cubical  and  rhombic, 
the  former  being  the  mineral  called  Senarmontite  and  the 
latter  Valentinite.  Silica,  SiO2,  is  known  in  four  different 
states.  When  fused  before  the  oxyhydrogen  blowpipe  it 
forms  an  amorphous  glass  of  sp.  gr.  2-22,  which  also  occurs 
in  the  minerals  opal  or  hyalite.  The  crystallised  varieties 
are — i.  Quartz,  which  is  hexagonal  (tetartohedral)  and  sp. 
gr.  2 '66  ;  2.  Tridymite,  also  hexagonal,  but  holohedral,  sp. 
gr.  2-3  ;  and  3.  Asmanite,  rhombic,  and  of  sp.  gr.  2*24. 

Titanic  acid,  TiO2,  is  another  example  of  a  trimorphous 
oxide;  the  commonest  variety,  Rutile,  belonging  to  the 
tetragonal  system,  has  the  fundamental  parameter  a  :  c=i 
:  0*6442,  and  sp.  gr.  4.25;  the  second  form.  Anatase,  is 
also  tetragonal,  but  the  crystals  are  of  pyramidal  habit, 
having  a  :  c=i  :  1778,  and  sp.  gr.  3-9.  The  third  form, 
known  as  Brookite,  is  rhombic,  and  has  the  sp.  gr.  4'i5. 
Carbonate  of  Calcium,  CaCO3  is  perhaps  the  most  familiar 
example  of  a  dimorphous  substance  being  rhombic,  and  of 
sp.  gr.  2  -9,  in  Aragonite,  and  rhombohedral  in  Calcite,  whose 
sp.  gr.  is  27.  The  cause  of  these  differences  in  form  was 


CHAP.  XVI.]  Isomorphism.  335 

for  a  long  time  sought  to  be  explained  by  slight  differences 
in  chemical  composition,  Aragonite  being  supposed  to  owe 
its  rhombic  character  to  the  presence  of  a  small  proportion 
of  carbonate  of  strontium ;  but  the  fact  of  their  essential 
similarity  in  composition  was  demonstrated  by  G.  Rose,  who 
proved  that  carbonate  of  calcium,  when  precipitated  from  a 
cold  solution,  consists  of  microscopic  rhombohedra  of  cal- 
cite  ;  but  when  the  precipitate  takes  place  from  a  boiling 
solution,  minute  prisms  of  arragonite  are  obtained.  The  ori- 
ginal establishment  of  the  phenomena  of  dimorphism  is  due  to 
Mitscherlich,  who  in  1823  first  demonstrated  the  differences 
between  the  form  of  native  sulphur  crystals  and  that  ob- 
tained artificially  by  crystallisation  from  fusion.  Subsequent 
researches  have  proved  that  numerous  other  substances 
possess  the  same  property,  and  their  number  may  be  con- 
siderably enlarged  when  the  comparison  is  extended  to 
substances  not  absolutely  identical  in  composition,  but  re- 
presented by  similar  formulae. 

Isomorphism. — The  fact  that  small  variations  of  the  pro- 
portions of  particular  components  are  possible  in  minerals 
without  changing  their  crystalline  form  was  known  to  the 
earlier  crystallographers,  and  an  explanation  was  pro- 
pounded by  Haiiy,  upon  the  assumption  of  the  dependence 
of  the  former  upon  the  quality  of  the  constituents  alone. 
Thus,  in  the  series  of  rhombohedral  carbonates  calcite,  mag- 
nesite,  dolomite,  siderite,  ankerite,  calamine,  &c.,  the  simi- 
larity in  crystallographic  characters  was  supposed  to  be 
caused  by  the  presence  of  a  small  proportion  of  calcium  in 
each  of  the  different  members,  which  exerted  a  dominating 
formative  influence  over  the  other  constituents,  and  gave  a 
general  resemblance  to  the  type  species  of  the  series  calcite 
or  carbonate  of  calcium.  This  view  was  shown  to  be 
erroneous  by  Mitscherlich,  who  demonstrated  in  1819  that 
the  relation  is  essentially  based  upon  quantity,  and  applied 
to  it  the  name  of  Isomorphism  (from  t<ros,  '  equal,'  and 
form).  Minerals  of  analogous  constitution,  that  is, 


336  Systematic  Mineralogy.          CHAP.  XVI. 

containing  the  same  number  of  atoms  of  like  kinds,  are 
isomorphous.  Thus  in  the  series  in  question,  that  of 
calcite,  the  whole  of  the  members  are  constituted  upon  the 
common  type,  RUCO3,  whence,  by  substituting  for  R"  equi- 
valents of  the  analogous  metals,  calcium,  magnesium,  iron, 
manganese,  and  zinc  successively  are  obtained — Calcite 
CaCO3,  Magnesite  MgCO3,  Siderite  FeCO3,  Manganese  spar 
MnCO3,  and  Calamine  ZnCO3 ;  and,  in  addition  to  these, 
there  are  others,  in  which  two  or  more  metals  are  present, 
such  as  Dolomite,  containing  both  calcium  and  magnesium; 
Ankerite,  calcium,  magnesium,  and  iron,  &c. — all  of  which 
are  rhombohedral  in  form  and  similar  in  constitution. 

In  the  strict  literal  sense  the  term  '  isomorphous '  is  only 
applicable  to  such  substances  as  are  cubic  in  crystallisation, 
as  in  the  other  systems,  the  relation  between  the  members  of 
isomorphous  series  is  one  of  close  analogy,  but  not  of 
identity.  Thus  in  the  above  series  the  polar  angle  of  the 
rhombohedron  varies  from  105°  5'  in  calcite  to  107°  40'  in 
calamine  ;  the  carbonates  of  iron,  manganese,  and  magne- 
sium giving  intermediate  values.  These  differences,  though 
small,  are  sufficient  to  establish  the  crystallographic  inde- 
pendence of  the  different  species,  and  therefore  the  term 
'  homeomorphous '  is  used  by  some  authors  to  express  the 
relation  of  such  minerals  ;  but  the  practice  is  not  followed 
to  any  very  great  extent 

In  addition  to  the  example  already  given,  the  following 
are  among  the  more  important  isomorphous  groups  : — 

Corundum  group.     Hexagonal  rhombohedraL 
Corundum  A1O3.      Hematite  F-eO3.     Chromic  oxide 

Apatite  group.     Hexagonal  pyramidal  hemihedral. 
Fluor-Apatite  3(Ca3P2O8)  CaFl2 
Chlor- Apatite  3(Ca3P2O8)  CaCl2 
Pyromorphite  3(Pb3P2O8)  PbCl2 
Mimetesite    .  3(Pb3As2O8)  PbCl2 
Vanadinite    .  3(Pb3V268)  PbCl2 


CHAP.  XVI.]  Isomorphism.  337 

These  occur  in  closely  allied  forms,  in  spite  of  the  quali- 
tative differences  in  composition  ;  the  isomorphous  relations 
of  these  elements  being  :—  calcium  and  lead  ;  phosphorus, 
arsenic  and  vanadium;  and  chlorine  and  fluorine. 

Spinel  group.     Cubic. 

Spinel  .  .  MgA*O4 
Magnetite  .  Fe¥eO4 
Chromite  .  Fe€K)4 
Franklinite  .  (ZnFeMn)  (£eM4i)O4 

Alum  group.     Cubic. 

Potash  alum      .    .     .     .  K2AtS4O16  24  Aq 
Ammonia  alum      .     .    .  (NH4)2A}S4O16 
Potash-iron  alum        .     .  K2FeS4O,6  24-Aq 
Ammonia-iron  alum       .  (NH4)2£eS4O16 
Potash-chrome  alum      .  K2G*S4O16  24A 
Ammonia-chrome  alum.  (NH4)2G*S4O16 


Only  the  first  two  members  of  the  above  series  occur  as 
natural  minerals,  the  others  having  been  prepared  arti- 
ficially. 

Most  of  the  isomorphous  groups  in  the  rhombic  system 
are  dimorphous  with  those  of  other  systems.  For  instance, 
the  rhombic  form  of  carbonate  of  calcium  or  Aragonite  is 
the  type  of  a  series  nearly  as  extensive  as  that  of  the  rhom- 
bohedral  form  of  the  same  substance,  or  Calcite,  including 
the  following  species  :  — 

Aragonite,  CaCO3  Strontianite,     .  SrCO3 

Witherite,  BaCO3  White-lead  ore,  PbCO3 

The  Diaspore  group,  also  rhombic  and   dimorphous  with 
that  of  Spinel,  includes  — 

Chrysoberyl,  BeMO4  Gothite,       H2¥e04 

Diaspore,      H2A4O4  Manganite, 


338  Systematic  Mineralogy.          [CHAP.  XVI. 

The  artificial  compound  produced  by  fusing  ferric  oxide 
and  lime  together,  described  by  Percy,  to  which  the  name 
Calciferrite  may  be  applied,  probably  belongs  to  this  series, 
being  very  similar  in  character  to  Gothite ;  its  crystallo- 
graphic  characters  have  not,  however,  been  exactly  deter- 
mined. 

By  an  extension  of  the  idea  of  isomorphism,  the  dimor- 
phism of  many  substances  may  be  indirectly  established. 
Thus  the  carbonates  of  the  type  RCO3  are  dimorphous  ;  but 
only  one  of  them,  carbonate  of  calcium,  is  actually  known  in 
both  systems  as  Aragonite  and  Calcite.  Isomorphous  mix- 
tures of  rhombohedral  carbonates,  as  might  be  expected, 
assume  the  calcite  form,  and  rhombic  ones  those  of  ara- 
gonite.  Carbonate  of  lead,  PbCO3  in  the  species  whitelead 
ore,  belongs  to  the  latter  group ;  but,  in  combination  with 
carbonate  of  calcium,  it  forms  Plumbo-calcite  (PbCa)CO3, 
which  is  rhombohedral,  and  cannot  therefore  be  supposed  to 
be  derived  from  the  aragonite  series  but  from  calcite  and  a 
dimorphous  rhombohedral  variety  of  carbonate  of  lead,  not 
known  independently. 

The  isomorphous  mixture  of  the  carbonates  of  calcium 

and  barium,  p^pQ3  f '  known  as  Alstonite,  has    the  same 

symmetry  as  its  constituents  Witherite  and  Aragonite  ;  but 
Baryto-Calcite,  which  is  of  similar  constitution,  belongs  to 
the  oblique  system,  proving  the  type  RCO3  to  be  actually 
trimorphous,  although  no  carbonate  of  a  single  base  is 
known  to  crystallise  in  the  latter  system.  Similar  cases  are 
presented  in  the  following  series,  which  establish  isomor- 
phous relations  between  the  dyad  sulphates  and  carbonates 
and  the  combination  of  both :  — 

Rhombohedrai.  Rhombic.  Oblique 

RCC-3     Calcite    CaCO3       Aragonite  CaCO3          Baryto-Calcite 

RSO4      Dreelite   j^f  °4  }     Anglesite      PbSO4         Glauberite 
LeadhilHte^SO.}      ^^ 


CHAP.  XVI.]  Poly  symmetry,  339 

Substances  of  the  above  kind  that  are  both  isomorphous 
and  heteromorphous  are  said  to  be  isodimorphous  or  isotri- 
morphous,  according  to  the  number  of  different  crystalline 
systems  in  which  they  occur. 

Poly  symmetry. — One  of  the  most  important  series  of  mine- 
rals, known  as  the  Hornblende- Augite  group,  is  represented  by 
the  general  formula  RnSiO3,  where  Rn=Ca,  Mg,  Fe,  or  Mn. 
The  members  of  this  group  are  not  isomorphous  in  the  sense 
of  having  the  same  crystallographic  symmetry,  as  some  of 
them  occur  in  the  rhombic,  others  in  the  oblique,  and  others 
in  the  triclinic  system  ;  but  a  general  similarity  in  the  geo- 
metrical elements  of  the  four  is  observed.  Thus,  Diopside 

or  Augite,  of  the  type  ^,  o-Q3 1  crystallising  in  the  oblique 

system,  has  the  fundamental  parameters  a  :  b  :  £=1-094  :  i  : 
0-591,  and  /3=74°  ;  while  those  of  Tremolite  or  Hornblende 

Mggio3  I  area  :  b  :  c—  0-544  :  i  :  0-294,  and /3= 75°  15'. 

The  parameters  of  the  axes  a  and  c  in  Augite  are  therefore 
approximately  double  those  of  Hornblende,  while  the  angle 
fl  is  nearly  the  same  in  both  species. 

,  ;;/MgSiO,  1  .   j  ,     ., 

A  second  group,      ^  &O-Q    >  >  represented  by  the  species 

Bronzite,  Hypersthine  and  Enstatite,  is  rhombic,  with  the 
parameters  a  :  b  :  c=  1*031  :  r  :  1*177.  These  may  be 
compared  with  those  of  Augite,  if  the  latter  be  referred  to 
a  system  of  axes  that  are  rectangular  or  nearly  so,  which  is 
done  by  considering  the  face  (ooi)  as  (101),  which  gives  the 
parameters  a  :  b  '.  c  =  i  052  :  i  :  0-296,  and  /3=89°4o'. 
If  the  same  face  be  further  noted  as  (104),  the  following 
close  approximation  between  these  new  oblique  parameters 
and  the  rhombic  ones  becomes  apparent : — 

b    :        c         /3 


Bronzite        .     i  '03 1 
Augite  .     1-052 


i    :    1-177.     9°° 
i    :    1-182.    89°  40'. 


340  Systematic  Mineralogy.         [CHAP.  xvi. 

Similar  approximations  between  the  elements  of  sub- 
stances of  analogous  or  identical  composition,  but  crystal 
Using  in  different  systems,  are  observed  in  the  rhombic 
and  hexagonal  varieties  of  Sulphate  of  Potassium  and  in 
Albite  and  Orthoclase.  They  are  included  by  Rammelsberg 
under  the  general  head  of  isomorphism,  but  the  special 
term  polysymmetry  has  been  applied  to  them  by  Scacchi. 

When  a  mineral  contains  both  dyad  and  hexad  bases,  it 
may,  by  the  progressive  substitution  of  one  metal  for  another 
of  the  same  class,  vary  considerably  both  in  composition 
and  physical  characters  without  change  of  form.  One  of 
the  best  examples  is  afforded  by  Garnet,  which  occurs  in 
many  varieties,  differing  considerably  both  as  regards  colour 
and  density,  but  all  crystallising  in  the  cubic  system — the 
rhombic  dodecahedron  being  the  dominant  form ;  the  ob- 
served range  of  the  four  principal  bases  being  as  follows  : 

Alumina  .  (A4O3)  o  to  22  per  cent. 

Ferric  Oxide  .  (F-eO3)  o  to  30        „ 

Lime        .  .  CaO     o  to  37         „ 

Magnesia  .  MgO    o  to  22         „ 

When,  however,  the  proportions  of  the  bases  in  a  lime- 
alumina  and  a  lime-iron  garnet  respectively  are  reduced  to 
the  above  values,  it  is  found  that  in  the  first  case  Ca  :  Ai= 
3  :  i,  and  A3  :  Si=i  :  3,  and  in  the  second  Ca :  Fe=3  :  i 
andFe  :  Si=i  :  3  ;  while  in  both  Ca  :  Si=i  :  i.  Whence  it 
appears  that  the  two  compounds  are  of  the  analogous  com- 
position— 

Ca3AiSi3O12and  Ca3FeSi3Oi2, 

and  that  those  containing  both  alumina  and  ferric  oxide  are 
isomorphous  mixtures  of  both  types  in  varying  proportions. 
In  addition  to  these,  other  varieties  are  known  contain- 
ing the  following  silicates — 

Mg3AtSi3O12  Mg3FeSi3O12 

Fe3A4Si3O12 

Mn3AiSi3O12 


CHAP,  xvi.]  Isomorphism.  341 

either  independently  or  in  combination  with  the  calcium 
silicates  given  above. 

The  term  Garnet,  therefore,  is  not  special  to  any  one  of 
these  compounds  in  particular,  but  distinguishes  a  group  of 
isomorphous  silicates,  which,  however  much  they  may  differ 
qualitatively,  have  the  above  ratio,  3  :  i  :  3,  for  their  dyad 
and  hexad  metals  and  silicon  respectively  common  to 
all,  or  may  be  represented  by  the  generalised  formula, 

II    VI 

R3ftSi3O12,  which  covers  every  possible  variety  of  compo- 
sition indicated  by  the  above  special  types. 

The  isomorphism  of  compounds,  not  containing  the 
same  number  of  elementary  atoms,  supposes  the  substi- 
tution of  the  elements  to  take  place  in  the  proportion  of 
their  equivalence,  two  atoms  of  a  monad  replacing  one  of 
a  dyad  element,  &c.  This  is  seen  in  the  Diaspore  group, 
where  H2  in  Gothite  and  Manganite  represents  Be  in  Chryso- 
beryl,  and  Fe,  Mn,  or  Zn,  in  the  analogous  dimorphous 
species,  Magnetite  and  Franklinite  of  the  spinel  series. 
Another  example  is  afforded  by  Oxygen  and  Fluorine,  O 
replacing  F12,  or  R20=RF1,  and  RO=RF12.  This  is  seen 

in  Topaz  \,   c^-pi5    \  ,  which  is  rhombic   and   isomorphic 

with  Andalusite  A4SiO5. 

The  isomorphism  of  analogous  compounds  of  monad 

I  II  VI 

(R),  dyad  (R),  and  hexad  (R)  elements  is  apparent  in  the 
Augite  group  of  silicates  which,  in  addition  to  the  varieties 
already  mentioned  as  represented  by  the  constitution  RSiO3, 
contains  others  both  in  the  augite  and  hornblende  series,  in 
whose  composition  sodium,  aluminium,  and  ferric  silicates 
form  part,  in  addition  to  the  dyad  metals  Ca,  Mg,  Fe,  &c. 

Of  these,  the  following,  Babingtonite  ^Jgj  Q     f  >  Achmite 


3Na2SiO3  1  Na?SiO3  1 

FeSiO3     V  ,  and  Aegirite  2  RSiO3      >  ,  appear  in  the  augite 
2£€Si3O9  j  FeSi3O9  J 


342  Systematic  Mineralogy.  [CHAP.  XVI. 

Na2SiO3  1 
form,  while  Arfwedsonite  RSiO3       >   assumes  that  of  horn- 


blende.  These  formulae  suppose  Na2SiO3  and  RSiO3  to 
be  equivalent  molecules,  three  of  which  correspond  to 
one  of  ferric  silicate  ¥eSi3O9.  In  one  instance,  in  the 

n 

augite  group,  the  R  elements  are  completely  absent :  this  is 

•zR  SiO     1  n  : 

in  Spodumene  ?AiSi  Q3  ['   where   R  is  rePlaced  by   2R 

i 
=Li,  Na,  and  ¥e  by  At     Here  2A1^  is  equivalent  to  3R. 

niv  vi 

The  isomorphism  of  RRO3  and  R2O3  is  illustrated  by 
the  case  of  Titanic  iron  ore,  a  term  applied  to  several 
minerals  of  varying  composition,  containing  Iron,  Titanium, 
and  Oxygen,  and  usually  some  Magnesium,  but  which,  ac- 
cording to  Rammelsberg,  can  be  represented  by  the  general 

expression  "L  fJ    3  \ ,  all  having  the  crystalline  form  of 

Hematite,  or  Fe2O3.  This  view  is  not  universally  adopted, 
as  another  hypothesis  supposes  them  to  contain  Ti2O3,  the 
blue  oxide  of  titanium.  The  aluminous  varieties  of  augite 

and  hornblende  may  be  similarly  represented  by  n .. ,  ^  3  I . 

A12U3    J 

The  most  remarkable  examples  of  isomorphism  combined 
with  dissimilarity  of  constitution  are  afforded  by  the  alkaline 
nitrates ;  potash-nitre,  or  saltpetre,  KNO3,  being  crystallo- 
graphically  almost  identical  with  Aragonite,  while  nitrate  of 
sodium,  NaNO3,  is  equally  close  in  form  to  Calcite. 

There  are  two  cases  of  isomorphism  of  minerals  not  of 
analogous  constitution  among  the  class  of  silicates.  These 
are  Spodumene  and  Petalite,  and  Anorthite  and  Albite. 
The  former  are  both  oblique  and  closely  allied  in  form,  but 
completely  dissimilar  in  composition — 

i 
Spodumene,  being     R6Al4Si13O45,  or  a  bisilicate  ;  and 

i 
Petalite  Rf)Al4Si30O75,  or  a  quadrisilicate. 


CHAP.  XVI.]  Growth  of  Crystals.  343 

Similarly,  in  the  second  case,  both  minerals  being  triclinic 
and  isomorphous  members  of  the  lime-soda  felspar  group — 

Anorthite,    CaA4Si2O8,  is  a  monosilicate     and 
Albite,          Na.,A4Si6O1G,  a  trisilicate. 

When  a  crystal  of  a  salt  is  placed  in  a  solution  of  some 
other  similar  salt  of  an  isomorphous  metal  brought  to  the 
crystallising  point,  it  will  increase  in  size  by  the  addition  of 
layers  of  the  new  salt,  which  will  be  symmetrically  disposed 
about  the  planes  of  the  nucleus,  exactly  in  the  same  manner 
as  would  have  happened  had  the  growth  been  contained  in 
the  original  solution.  The  form  will  therefore  be  preserved, 
but  the  crystal  will  obviously  be  only  a  mixture  of  hetero- 
geneous substances,  and  its  composite  nature  will  be  appa- 
rent if  there  is  any  marked  difference  in  physical  characters 
between  the  different  constituents.  One  of  the  best  examples 
of  this  kind  of  structure  is  furnished  by  the  double  sulphates 

I     VI  I 

of  the  alum  series  R2R2S4O1624Aq,  where  R2  may  be  either 

VI 

ammonium,  potassium,  or  some  other  monad  metal,  and  R2 
either  Chromium,  Iron,  or  Aluminium.  Two  of  these,  the 
ammonia-aluminium  and  potassium-aluminium  salts  are 
colourless,  while  the  chromium  and  iron  salts  are  strongly 
coloured,  the  former  being  dark  green  and  the  latter  violet, 
so  that  crystals  formed  from  the  solutions  of  two  or  more  of 
them  present  strongly  contrasted  alternating  bands  of  colour 
upon  a  cross  section  ;  or,  if  one  of  the  colourless  salts  is  from 
the  outer  layer,  they  may  appear  as  transparent  octahedra  with 
coloured  centres.  Crystals  of  this  kind,  although  illus- 
trating the  phenomena  of  isomorphism  in  a  graphic  manner, 
are  obviously  only  mechanical  mixtures  whose  heterogeneous 
character  is  plainly  visible,  and  cannot  therefore  be  repre- 
sented as  compounds  of  isomorphous  bases  in  the  same 
sense  as  those  of  dolomite,  pearl  spar,  and  other  minerals 
are,  where  the  combination  extends  to  the  individual  crystal- 
line molecules.  They  are,  however,  of  considerable  import- 


344  Systematic  Mineralogy.         [CHAP.  XVI. 

ance  as  illustrations  of  facts  which  occur  in  nature  tolerably 
frequently.  Thus  crystals  of  Vanadinite,  3PbV2O8  PbCl2, 
from  Russia,  are  occasionally  found  to  contain  a  nucleus 
of  the  isomorphous  species  Pyromorphite,  3PbP2O8PbCl2, 
which  is  of  the  same  hexagonal  form.  Crystals  of  Tour- 
maline when  transparent  are  also  commonly  observed  to 
be  banded  in.  different  colours  which  correspond  to  differ- 

ii 

ences  of  composition  in  the  R  bases.  In  the  felspar  group, 
apparently  homogeneous  crystals  are  often  made  of  alter- 
nations of  the  isomorphous  minerals  Albite  and  Orthoclase, 
and  these  being  colourless,  it  is  often  difficult  to  distinguish 
one  from  the  other,  the  use  of  optical  tests  being  necessary 
in  such  cases.  The  same  thing  probably  occurs  in'  many 
other  minerals,  and  is  the  cause  of  the  discrepancies  between 
the  theoretical  composition  as  required  by  the  formula,  and 
the  results  obtained  by  analysis.  In  this  respect  minerals 
differ  essentially  from  crystallised  salts  artificially  prepared, 
which  may,  by  particular  manipulation,  be  obtained  in  a 
state  of  almost  absolute  purity,  while  the  former  almost 
invariably  contain  some  matters  foreign  to  their  essential 
constituents. 

Another  interesting  case  of  crystals  made  up  of  alter- 
nating layers  of  isomorphous  compounds  of  different  com- 
position is  occasionally  seen  in  the  arsenides  of  Nickel  and 
Cobalt  (NiAs2,  CoAs2).  These  are  both  cubical,  and  found 
in  large  lead-grey  crystals  apparently  perfectly  uniform  in 
composition,  but,  when  exposed  to  damp  air,  become  oxi- 
dised with  the  formation  of  basic  arseniates  of  the  respective 
metals — that  of  cobalt  being  pink  and  that  of  nickel  pale 
green,  so  that  the  crystals  when  broken  across  often  weather 
in  layers  which  are  alternately  coated  with  pink  and  green 
incrustations,  according  as  one  or  other  metal  predominates 
in  the  particular  layer. 

It  is  probable  that  the  universal  presence  of  gold  in 
minute  quantities  in  such  minerals  as  galena,  PbS,  and  iron 


CHAP.  XVII.]         Alteration  of  Minerals.  345 

pyrites,  FeS2,  may  be  due  to  a  mechanical  isomorphous 
intermixture  of  this  kind,  as  all  these  species  are  cubical  in 
form,  and  there  is  no  reason  to  suppose  that  the  gold  is  in 
chemical  combination  as  it  may  often  be  extracted  by  the 
process  of  solution  in  mercury  known  as  amalgamation. 


CHAPTER  XVII. 

ASSOCIATION   AND   DISTRIBUTION   OF   MINERALS. 

MINERALS,  when  exposed  to  the  action  of  air,  water,  car- 
bonic acid,  and  other  agents  of  a  similar  kind  tending  to 
produce  alteration  in  chemical  composition,  show  very 
unequal  degrees  of  stability ;  some  species,  such  as  gold, 
diamond,  and  graphite,  the  different  forms  of  carbon,  quartz 
and  tin  ore  being  practically  unalterable,  as  they  are  neither 
susceptible  of  change  by  oxidation,  nor  reduction  in  air  at 
the  ordinary  temperature,  and  almost,  if  not  quite,  insoluble 
in  meteoric  or  ordinary  spring  waters ;  while,  on  the  other 
hand,  soluble  and  hydrated  salts,  especially  those  containing 
the  alkaline  metals,  and  the  dyad  forms,  iron,  manganese,  and 
calcium,  are,  in  a  greater  or  less  degree,  liable  to  change  either 
in  form  or  composition  under  ordinary  atmospheric  vicissi- 
tudes. The  following  are  some  of  the  most  genera1  cases  : — 
i.  Alteration  by  loss  of  water.  This  is  called  efflores- 
cence, and  is  characteristic  of  minerals  containing  water  of 
crystallisation  which  may  in  dry  air  be  given  off  either  entirely 

or  in  part.    Crystals  of  Laumonite,  \  VfJ-  A    >  +4Aq,  lose 

t  Al2ol3U9  J 

water  by  exposure  and  fall  to  pieces,  although  the  change 
may  be  very  gradually  effected. 

Crystals  of  gypsum,  CaSO42Aq,  which  are  perfectly 
transparent  when  fresh,  are  often  found  in  dry  countries  to 
become  opaque  either  wholly  or  in  part  from  a  partial  dehy- 
dration when  exposed  to  the  air. 


346  Systematic  Mineralogy.        [CHAP.  XVII. 

2.  Solution  and  absorption  of  water.     Minerals  whose 
crystals  lose  their  form  by  absorption  of  atmospheric  mois- 
ture, and  are  ultimately  converted  into  solutions  are  said  to- 
be  deliquescent;  this  property  is  common  to  many  of  the 
soluble  alkaline  salts,  such  as  nitrate  of  soda,  common  salt, 
sal  ammoniac,  &c. 

3.  Change  by  oxidation  of  one  or  more  constituents.     This 
is  a  very  common  occurrence  in  ferrous  or  manganous  com- 
pounds, and  is  generally  known  as  rusting.     It  is   most 
rapidly  developed  in  the  soluble  salts  of  these  metals.    Thus, 

n 

Ferrous  sulphate  (FeSO4yAq)  is  of  a  well-defined  consti- 
tution and  form,  but  the  crystals  can  only  be  preserved  in 
absolutely  dry  air,  or  in  the  vapour  of  a  hydrocarbon,  as 
under  ordinary  conditions  of  exposure  they  become  dull 
and  rusted  through  the  production  of  ferric  salts,  a  very  large 
series  of  which  are  known  in  nature,  and  are  produced  by 
progressive  oxidation,  the  ultimate  product  of  such  alter- 
ation being  a  Ferric  hydrate  (H6Fe4O9)  and  acid  ferric 
sulphate.  In  the  same  way,  carbonates  containing  the  same 
base,  such  as  Siderite,  Pearl  spar,  Dolomite,  become  inva- 
riably rusted  externally  when  exposed  to  the  air,  even  when 
the  proportion  of  iron  present  is  but  small.  Manganous 
compounds  which,  when  fresh,  are  of  a  delicate  rose  red, 
are  even  more  susceptible,  as  they  turn  brown  by  exposure 
to  sunlight,  and  ultimately  become  brown  or  black  by  the 
formation  of  manganic-oxide  MnO2,  the  brown  oxide  Mn3O4, 
or  their  hydrates.  For  this  reason,  specimens  of  minerals 
such  as  Rhodonite  MnSiO3  and  Diallogite  MnSO4  are  gene- 
rally kept  in  the  dark,  or  the  cases  containing  them  in 
museums  are  screened  from  direct  light. 

Most  metallic  sulphides  and  arsenides  are  similarly  liable 
to  change  by  oxidation  in  damp  air,  with  the  formation  of 
oxysalts  of  their  constituents.  Thus,  Galena  PbS  gives 
rise  to  Anglesite  PbSO4,  Zinc  blende  ZnS  to  Zinc  vitriol 
ZnSO47Aq,  Cobalt  speiss  CoAs2  to  the  hydrated  arseniate 


CHAP.  XVII.]   Action  of  Air  and  Carbonic  Acid.        347 

Co3As2O8 .  +8Aq,  known  as  Cobalt  bloom,  and  Nickel 
speiss  NiAs2  to  the  corresponding  Nickel  salt  or  nickel  bloom. 
Bisulphide  of  iron  FeS2,  which  is  among  the  commonest  of 
minerals,  forming  the  dimorphous*  species  Pyrites  (cubical), 
and  Marcasite  (rhombic),  besides  being  found  in  various 
isomorphous  mixtures  with  other  metallic  sulphides  and 
arsenides,  yields,  by  the  simultaneous  oxidation  of  both  con- 
stituents, Ferrous  sulphate  or  Green  vitriol  FeSO4,  and 
Sulphuric  acid,  the  former  salt  going  through  the  changes 
previously  noticed,  the  ultimate  product  being  Ferric  hydrate 
and  basic  ferric  sulphate ;  but  when  aluminium  or  sodium 
compounds  are  within  reach,  the  iron  salts  may  be  com- 
pletely destroyed  with  the  formation  of  the  sulphates  of 
these  metals,  Glauberite,  Gypsum,  and  Alum.  This  group 
of  reactions  is  one  of  the  most  important  in  the  whole  range 
of  natural  chemistry,  as  it  is  concerned  in  the  production  of 
the  quantity  of  soluble  sulphates  present  in  most  terrestrial 
waters,  and  which,  in  the  case  of  mineral  springs,  often 
amounts  to  a  considerable  percentage.  This  change,  which 
is  generally  known  as  vitriolescence,  goes  on  more  rapidly 
in  the  rhombic  and  granular  varieties  of  pyrites  than  in  the 
crystallised  cubes ;  and  when  specimens  of  such  minerals 
are  kept  in  cabinets  they  often  develop  the  unpleasant  pro- 
perty of '  eating  up  their  labels,'  that  is,  the  labels  are  rotted 
and  destroyed  by  the  sulphuric  acid  formed,  the  change 
being  accompanied  with  the  formation  of  capillary  crystals 
of  ferrous  sulphate. 

Sulphuric  acid  is  also  formed  by  the  oxidation  of  sul- 
phurous acid  in  the  steam  jets  of  volcanoes  or  fumaroles. 
In  such  localities  it  is  common  to  find  the  felspathic  com- 
ponent of  the  rocks  reduced  to  a  mass  of  clay  variegated 
with  parti-coloured  patches,  which  are  essentially  alum  and 
ferric  sulphates. 

4.  Change  by  action  of  carbonic  acid.  All  natural  waters, 
whether  terrestrial  or  atmospheric,  hold  more  or  less  car- 
bonic acid  in  solution,  and,  as  such,  are  a  cause  of  alteration, 


348  Systematic  Mineralogy.        [CHAP.  XVII. 

which,  though  less  energetic  than  sulphuric  acid  in  any 
special  case,  is,  as  a  whole,  more  important  from  the  univer- 
sality of  its  action.  The  effects  produced  are  of  two  kinds  : 
ist,  the  solution  of  carbonate  of  calcium,  magnesium,  and 
analogous  metals,  and  of  their  phosphates  and  fluorides,  is 
promoted,  as  these  salts  are  not  sensibly  soluble  in  pure  water, 
but  dissolve  more  or  less  readily  in  water  saturated  with 
carbonic  acid  ;  and  2nd,  the  double  silicates  containing 
alkaline  metals  and  aluminium,  typified  by  the  felspar  group, 
are  attacked  and  decomposed  with  the  separation  of  the 
alkaline  silicate,  which  dissolves  and  is  ultimately  decom- 
posed with  the  production  of  alkaline  carbonates  and  soluble 
silica,  while  the  insoluble  aluminium  silicate  becomes  hy- 
drated,  forming  the  mineral  Kaolin  or  China  clay.  This 
change,  known  as  kaolinisation,  affects  many  minerals  besides 
felspars,  and  is  probably  concerned  in  the  production  of 
most  clay  deposits.  Silicate  of  lime  CaSiO3  is  also  decom- 
posed by  carbonic  acid  water,  as  are  also  ferrous  silicates, 
but  less  readily  than  the  former.  Waters  containing  alkaline 
carbonates,  especially  when  concentrated  as  in  hot  springs, 
also  have  a  marked  solvent  effect  upon  silica,  which  is  after- 
wards deposited  as  opal,  hyalite,  or  chalcedony,  or  even 
quartz.  The  sulphides  of  the  heavy  metals,  lead,  zinc,  or 
copper,  are  slowly  converted  into  carbonates  under  the  action 
of  atmospheric  waters  containing  carbonic  acid,  and  are 
therefore  commonly  found  to  have  been  so  changed  in 
mineral  veins  at  or  near  the  surface. 

5.  CJiange  by  reducing  agents. — These  are,  to  some  ex- 
tent, inverse  reactions  to  those  involving  oxidation.  Ferrous 
sulphate,  when  kept  in  contact  with  decomposing  organic 
matter,  whether  animal  or  vegetable,  is  converted  into 
sulphide,  crystals  of  iron  pyrites  having  been  produced  in 
this  way  both  naturally  and  artificially.  Hydrated  ferric 
oxide  is  in  like  manner  reduced  by  decaying  vegetable 
substances  and  carbonic  acid,  producing  ferrous  carbonate. 
Water  containing  alkaline  sulphates  may,  in  contact  with 


CHAP.  XVII.]  Pseudomorphism.  349 

organic  matter,  produce  sulphides  of  copper,  lead,  &c.,  by 
direct  action  upon  those  metals,  as  is  shown  by  Daubre'e  to 
have  taken  place  in  the  basins  of  certain  thermal  alkaline 
springs  at  Plombieres,  where  copper  pyrites  and  antimonial 
grey  copper  ore  have  been  produced  from  Roman  coins  im- 
bedded in  the  mud  of  the  spring. 

6.  Alteration  by  chlorides. — The  mineral  Atacamite,   or 

oxychloride  of  copper,  TT^Q2  >,  is   a   tolerably  common 

product  of  the  alteration  of  copper  pyrites  in  the  mining 
districts  of  Chili,  Peru,  and  South  Australia.  It  is  readily 
produced  when  sulphuretted  copper  ores  are  exposed  to  the 
joint  action  of  air  and  sea-water,  and  is  probably  due  to  a 
similar  action  in  the  mines  in  question  which  are  situated 
in  hot,  dry  countries,  with  little  or  no  rainfall,  and  where,  in 
consequence,  the  alkaline  chlorides  in  the  rocks  have  not 
been  completely  washed  out.  Probably  most  of  the  chloride 
of  silver  found  in  mineral  veins  is  to  be  attributed  to  the 
same  cause  as  a  product  of  the  alteration  of  sulphide  of 
silver. 

Evidence  of  alteration.  —  Pseudomorphism. — The  trans- 
formation of  a  mineral  by  any  of  the  methods  previously 
described  may  be  more  or  less  evident,  according  to  the 
nature  of  the  change  and  the  extent  to  which  it  has  pro- 
gressed. The  earliest  stages  of  alteration  are  marked  prin- 
cipally by  change  of  colour  or  lustre  in  the  faces  of  the 
crystals,  which  become  superficially  altered  while  preserving 
their  original  structure  and  composition  within  ;  while,  on 
the  other  hand,  the  alteration  may  be  so  great,  and  the  de- 
velopment of  new  minerals  so  completely  effected,  that  the 
derivation  of  the  latter  can  only  be  inferred  by  generalisation 
upon  evidence  obtained  in  other  cases.  Such  evidence  may 
in  some  cases  be  obtained  by  direct  experiment ;  but  in  the 
larger  number  of  instances  it  is  furnished  by  what  are  known 
as  pseudomorphs,  i.e.  minerals  that  appear  in  crystalline 
forms  not  compatible  with  their  chemical  constitution,  and 


350  Systematic  Mineralogy.        [CHAP.  XVII. 

must  therefore  have  been  altered  by  a  partial  or  complete 
modification  of  their  constituent  elements,  while  retaining 
the  forms  proper  to  their  original  composition.  The  study 
of  this  most  interesting  branch  of  mineralogy  has  been 
systematised  by  Haidinger,  Blum,  Volger,  and  other  ob- 
servers, and  the  observed  cases  have  been  classified  under 
the  following  heads  : — 

1.  Pseudomorphism  by  substitution. — This  implies  a  gra- 
dual replacement  of  the  original  substance  by  another,  by 
means  of  simultaneous   solution  and   deposition,  without 
involving  chemical  action.    The  pseudomorphs  of  Quartz, 
or  other  varieties  of  Silica,  after  Calcite,  Fluor  Spar,  Barytes, 
and  similar  minerals,  are  of  this  kind  :  the  siliceous  matter 
having  been  deposited  from  solution,  part  passu,  as  the  ori- 
ginal crystal  was  attacked  and  dissolved ;   the  general  order 
observed  in  such  cases  being  that  the  replacing  substance  is 
less  soluble  than  that  forming  the  original  crystal.     The 
fossilisation  of  the  remains  of  plants  and  animals  by  Silica 
and  other  minerals  is  also  to  be  referred  to  this  kind  of  action. 

2.  Pseudomorphism   by  incrustation. — In  this  case  one 
mineral  having  been  deposited  upon  another,  and  the  older 
one  subsequently  removed,  evidence  of  former  existence  of 
the  latter  is  supplied  by  the  hollow  impression  of  its  crystals 
retained  in  the  second  or  incrusting  species.     As  instances 
of  this  kind  may  be  mentioned  :  Quartz  upon  Fluor  Spar, 
Iron  Pyrites  upon  Barytes,  Quartz  upon  Barytes,  Chlorite 
upon  Dolomite,  and  Siderite  upon  Barytes.     In  all  these 
instances,  the  free  or  outer  surface  of  the  incrusting  mineral 
is  generally  developed  according  to  its  own  form,  while  the 
under  side  of  the  layer  forms  an  empty  mould  of  the  crystal 
of  the  mineral  upon  which  it  was  originally  deposited,  but 
which  has  been  since  removed.     It  often  happens,  however, 
that  the  hollow  space  so  produced  is  subsequently  filled 
either  with  the  incrusting  substance  or  some  third  mineral, 
with  the  production  of  a  substitution-pseudomorph  of  a  more 
complex  kind  than  those  of  the  previous  case. 


CHAP.  XVII.]  Paramorphism.  351 

3.  Pseudomorphism  by  alteration.  —  Under  this  general 
head  are  included  the  following  particular  cases  :  — 

(a)  By  loss  or  diminution  of  constituents. 

(£)  By  gain  or  increase  of  one  or  more  constituents. 

(c)  By  interchange  or  substitution  of  one  or  more  con- 
stituents. 

The  first  of  these  cases  is  exemplified  by  pseudomorphs 
of  Anhydrite  after  Gypsum,  where  the  change  is  a  simple 

dehydration  ;  Calcite  after  Gaylussite  <  rvrv}  3    f  >  where 

the  carbonate  of  sodium  "is  removed;  and  native  copper 
after  Cuprite  (Cu2O),  where  there  is  reduction  or  removal 
of  oxygen. 

The  following  are  examples  of  the  second  case  :  Gypsum 
after  Anhydrite,  involving  the  addition  of  water  ;  Malachite 

after  Cuprite  (Cu2O),  by  addition  of  oxygen, 


carbonic  acid,  and  water  ;  and  Anglesite  (PbSO4)  after 
Galena  (PbS),  by  simultaneous  oxidation  of  both  lead  and 
sulphur. 

The  third  and  last  case  is  wider  in  scope  than  either  of 
the  preceding,  and  includes  the  larger  number  of'  observed 
pseudomorphs,  some  of  the  most  prominent  being  the 
following  :  Limonite  (H6Fe4O9)  after  Iron  Pyrites  (FeS2), 
and  Siderite  (FeCO3),  by  loss  of  sulphur  and  addition  of 
water  in  both  instances  ;  White  Lead  ore  (PbCO3)  after 

Galena  (PbS),  by  loss  of  sulphur  and  gain  of  carbonic  acid  ; 

i 
and  Kaolin  (Al2Si2O72Aq)  after  Felspar  (R2Al2Si6Olfi),  by 

the  loss  of  an  alkaline  silicate  and  the  addition  of  water. 

The  term  paramorphism  is  applied  to  a  particular  class 
of  pseudomorphism  where  a  mineral  occurs  in  a  form 
proper  to  its  composition,  but  having  the  structure  proper 
to  a  dimorphous  mineral  of  the  same  composition.  Ex- 
amples of  this  are  furnished  by  the  change  of  oblique, 
prismatic  crystals  of  sulphur  into  an  aggregate  of  rhombic 
crystals  without  change  of  exterior  form,  aragonite  with 


252  Systematic  Mineralogy.          [CHAP.  xvn. 

calcite  structure,  and  augite  with  hornblende  structure  in 
Uralite. 

The  derivative  character  of  pseudomorphs  is  evident 
from  the  absence  of  the  structural  peculiarities,  such  as 
cleavage,  lustre,  &c.,  proper  to  the  forms  imitated,  and  as  a 
rule  they  are  dull,  made  up  of  amorphous  material,  very 
generally  hydrated,  such  as  Kaolin  and  Steatite,  dull  or 
waxy  on  a  fractured  surface,  and  usually  spongy  or  hollow, 
especially  when  produced  by  total  replacement;  but  in 
some  cases  of  pseudomorphism  by  alteration,  the  change 
may  be  effected  without  alteration  of  volume,  so  that  the 
pseudomorph  may  be  as  compact  as  the  original  crystal. 
This  is  especially  well  seen  in  the  common  case  of  Limonite 
pseudomorphs  after  cubes  of  Iron  Pyrites,  where  the  latter, 
or  original,  substance  is  completely  transformed  into  a  com- 
pact mass  of  the  former,  while  preserving  all  the  external 
characters  of  the  crystals,  even  to  the  twin  striations  upon 
the  faces.  Here  the  proportion  of  the  unaltered  constituent 
Iron  (467  per  cent.)  in  the  molecule  of  Pyrites,  is  to  that 
in  the  molecule  of  Limonite  (60  per  cent.)  as  i  to  1-3, 
while  their  specific  gravities  are  in  the  inverse  ratio  of  1-4 
to  i,  or  5*0  for  Iron  Pyrites  and  3*6  for  Limonite. 

In  the  parallel  case  of  pseudomorphism  of  Limonite 
after  Siderite,  the  change  is  attended  with  diminution  of 
volume,  as  both  are  of  nearly  the  same  density,  3-5,  but 
Siderite  contains  only  45  per  cent,  of  iron  against  60  per 
cent,  in  Limonite,  so  that  the  volume  of  the  pseudo- 
morphs can  only  be  about  |ths  of  that  of  the  original  crystal. 
Actually,  however,  the  volume  is  considerably  less,  as 
Siderite  invariably  contains  more  or  less  of  the  isomorphous 
bases — Lime,  Magnesia,  and  Manganous  Oxide,  which  are 
either  removed  in  solution  as  carbonates  or  separate  as 
Pyrolusite,  or  other  manganese  ores,  by  oxidation.  These 
differences  do,  however,  actually  correspond  to  structural 
differences  in  the  resulting  minerals  on  the  large  scale,  as 
brown  Iron  Ores  (Limonite)  produced  by  the  alteration  of 


CHAP.  XVII.]  Origin  of  Minerals.  .355 

\f~ 

masses  of  Pyrites  are  usually  dense  or  compact  in  s, 
while  those  produced   similarly  from   Spathic   Carbo. 
(Siderite)  are,  as  a  rule,  spongy  or  cellular. 

In  many  cases  the  alteration  of  minerals  is  attended  with 
very  considerable  increase  of  volume,  as,  for  example,  in  the 
conversion  of  metallic  Copper  into  Malachite,  or  Iron  into 
Limonite.  In  the  latter  instance  the  increase  of  volume  is 
from  eight-  to  tenfold,  so  that  the  change  may  be  attended 
with  considerable  mechanical  action  when  effected  in  a  con- 
fined space.  Numerous  examples  of  this  action  may  be 
seen  in  old  wrought-iron  work,  which  has  been  long  exposed 
to  the  weather,  where  rusting  has  gone  on  between  surfaces 
originally  in  contact,  but  which  have  been  thrust  apart  by 
the  rust  formed  between  them.  An  analogous  action  may 
be  assumed  as  taking  place  in  the  change  of  complex 
silicates  into  alkaline  carbonates,  silica  and  clay,  owing  to 
the  greatly  increased  volume  of  these  products  as  compared 
with  that  of  the  original  mineral. 

Origin  of  minerals.  Questions  as  to  the  probable  origin 
and  method  of  formation  of  minerals  are  among  the  most 
interesting  in  the  whole  field  of  mineralogy,  but  the  material 
available  for  their  solution  is  comparatively  small.  In  many 
instances  the  evidence  of  pseudomorphs  is  sufficient  to  show 
a  secondary  origin,  or  transformation  from  a  pre-existing 
combination.  The  researches  of  Daubree  upon  deposits 
formed  in  the  conduits  of  the  thermal  springs  supplying 
mineral  baths  in  France  and  Algiers,  which  have  been  in 
use  since  the  Roman  conquest  of  Gaul,  have  shown  conclu- 
sively that  minerals  of  the  class  of  Zeolites  may  be  readily 
produced  by  the  action  of  slightly  alkaline  heated  waters 
upon  the  rocks  they  traverse  when  continued  for  a  period 
of  many  centuries ;  and  similarly,  the  condition  of  the 
bronze  objects  found  in  the  remains  of  Assyrian  and  other 
ancient  cities,  by  their  conversion  into  Ruby  Copper  ore  and 
Malachite  afford  not  only  a  proof  of  the  essentially  secondary 
character  of  these  minerals,  if  it  be  needed,  but  also,  in 

A  A 


Systematic  Mineralogy,        [CHAP.  xvn. 

some  degree,  a  measure  of  the  time  required  for  their  for- 
mation. 

With  minerals  of  a  more  complex  character,  especi- 
ally among  Silicates,  direct  evidence  can  rarely  be  obtained, 
and  in  such  cases,  therefore,  the  undesigned  production  of 
compounds  similar  in  form  and  composition  to  natural 
minerals  in  slags  and  other  furnace  products,  or  what  are 
usually  known  as  artificial  minerals,  is  of  special  significance. 

The  following  are  amongst  the  minerals  that  have  been 
most  frequently  observed  in  furnace  products  definitely  crys- 
tallised :— 

LimeAugite  (CaSiO3)  in  the  slags  of  several  blast  fur- 
naces smelting  iron  ore.  Manganese  Augite  (MnSiO3),  or 
approximating  to  that  composition,  which  resembles  the 
natural  mineral  Babingtonite,  but  is  not  exactly  similar.  This 
is  found  in  slags  produced  in  the  Bessemer  process  of  steel- 
making.  Potash  Felspar  (KAiSi3Oi2),  in  minute  crystals  in 
the  walls  of  furnaces  smelting  the  copper-schist  of  Mansfeld, 
exactly  similar  to  the  natural  mineral,  but  of  rare  occur- 
rence. Humboldtilite,  in  modified  square  prisms,  very 
much  larger  than  those  of  the  natural  mineral  which  occurs 
in  the  lavas  of  Vesuvius:  these  are  commonly  seen  in  the 
older  blast-furnace  slags  of  South  Staffordshire.  Iron  Chry- 
solite, or  olivine  (Fe2SiO4),  corresponding  in  composition 
to  the  doubtful  mineral  species  Fayalite.  This  occurs  very 
commonly  in  the  slags  of  puddling  furnaces,  and  also  at 
times  in  those  produced  in  smelting  lead  or  copper  ores,  or 
generally  where  slags  that  are  essentially  ferrous  silicates  are 
formed.  The  crystals  have  the  form  of  the  isomorphous 
mineral  chrysolite  or  olivine  (Mg2SiO4),  but  the  latter  is  not 
found  in  furnace  products,  for  the  obvious  reason  that  slags 
consisting  mainly  of  magnesian  silicates  are  not  susceptible 
of  formation  under  the  ordinary  conditions  of  working,  owing 
to  the  refractory  character  of  such  compounds,  and  the  use 
of  magnesia  in  fluxes  is  therefore  carefully  avoided.  Mag- 
netite (Fe3O4)  is  common  in  octahedral  crystals  in  the  slags 


CHAP.  XVII.]  Artificial  Minerals.  355 

produced  in  the  later  stages  of  the  puddling  process,  when 
the  amount  of  iron  taken  up  is  in  excess  of  that  required 
to  form  a  definite  silicate  with  the  silica  present.  It  is 
also  produced  when  steam  is  passed  at  a  red  heat  over 
ferrous  sulphide,  which  has  probably  been  the  mode  of 
formation  of  the  brilliant,  artificial  crystals  found  occa- 
sionally in  the  deposits  formed  in  furnaces  smelting  pyritic 
silver  ores  at  Freiberg. 

Galena,  PbS,  in  brilliant,  cubical  crystals  and  columnar 
aggregates,  is  tolerably  common  in  deposits  apparently  the  re- 
sult of  sublimation  in  the  throats  of  blast  furnaces  smelting 
lead  ore,  and  where  the  ore  contains  zinc,  Blende  (sulphide 
of  zinc)  and  oxide  of  zinc  (the  latter  not  occurring  as  a  natural 
mineral)  may  be  deposited  in  a  similar  manner. 

When  metallic  sulphides  are  roasted  by  burning  in 
heaps  in  the  open  air,  numerous  minerals  are  formed  by 
partial  oxidation  of  the  more  volatile  constituents,  especially 
sulphur  and  arsenic,  and  deposit  in  the  cooler  portions  of 
the  heap  in  a  manner  analogous  to  that  observed  in  solfataras 
and  other  volcanic  emanations.  Among  these  are  SulpJmr, 
Arsenwusaa'd(As2O2),  tfaztgar (AsS),  and  Orpiment  (As2S3). 
This  association  of  realgar  crystals  with  sulphur  is  common 
at  the  solfatara  of  Naples.  Sal  Ammoniac  (NH4C1)  is 
occasionally  deposited  in  the  same  manner  upon  waste  heaps 
over  burning  coal  slack.  Specular  hematite  (¥-eO3)  in  minute, 
brilliant  crystals,  is  occasionally  found  on  surfaces  of  salt- 
glazed  pottery  (such  as  drain-pipes,  &c.) ;  these  are  a  conse- 
quence of  the  action  of  steam  upon  ferric  chloride  (Fe2Cl6), 
which  results  in  the  production  of  ferric  oxide  and  hydro- 
chloric acid.  In  this  process,  common  salt  is  thrown  into  a 
kiln  when  the  clay  goods  are  brought  up  to  a  bright  red  heat, 
and  in  the  presence  of  water  vapour  is  decomposed  with  the 
formation  of  a  glaze  (silicate  of  soda)  upon  the  heated  surface 
of  the  ware,  hydrochloric  acid  and  some  ferric  chloride  from 
the  iron  contained  in  the  clay  being  volatilised.  This  reaction 
explains  the  formation  of  the  veiy  brilliant  crystals  of  specular 

A  A  2 


356  Systematic  Mineralogy.       [CHAP.  XVII. 

iron  ore  found  upon  the  lavas  of  Vesuvius,  Ascension  Island, 
and  other  volcanic  centres,  hydrochloric  acid  and  ferric 
chloride  being  commonly  found  in  the  steam  emitted  from 
fumaroles  during  and  after  periods  of  eruption. 

Graphite  in  the  form  of  crystalline  scales,  and  occasion- 
ally in  masses  of  considerable  size,  and  closely  resembling 
the  natural  mineral,  is  a  common  product  of  blast  furnaces 
smelting  iron  ores,  but  the  method  of  its  formation,  namely, 
separation  from  solution  in  molten  cast  iron,  cannot  be  con- 
sidered as  analogous  to  any  process  likely  to  produce  it  in 
nature. 

Water  at  high  temperatures,  when  it  is  made  to  act  under 
conditions  by  which  the  formation  of  vapour  is  prevented,  or 
when  under  considerable  pressure,  has  a  powerful  s'olvent 
action  upon  many  substances  which  are  not  affected  by  it 
under  ordinary  temperatures  and  pressures,  and  may  in  such 
a  case  give  rise  to  considerable  danger  to  substances  sub- 
mitted to  its  action.  The  principal  experiments  upon  this 
point  are  due  to  Daubree,  who  found  that  hard  glass,  a 
homogeneous  silicate  of  lime  and  soda,  may  be  converted 
into  crystallised  quartz  and  pyroxene  at  a  comparatively  low 
temperature  in  this  way.  It  is  probable  that  an  action  of 
this  kind  may  be  concerned  in  the  production  of  crystalline 
rocks  containing  quartz,  with  orthoclase  and  other  silicates, 
by  the  slow  rearrangement  of  masses  originally  homogeneous, 
when  solidified  by  the  process  which  is  generally  known  as 
devitrification,  or  the  transformation  of  a  glassy  substance 
into  an  opaque  mass  containing  crystals. 

Another  agent  of  considerable  importance  in  the  produc- 
tion of  minerals  is  probably  boracic  acid,  which,  though 
practically  fixed  when  exposed  alone  to  a  very  high  tempera- 
ture, is  sensibly  volatile  even  at  the  boiling  point  of  water  in 
an  atmosphere  of  steam.  The  method  of  occurrence  of 
minerals  containing  borax,  especially  tourmaline,  in  veins 
in  granite  and  other  rocks,  seems  to  indicate  a  formation  by 
a  process  analogous  to  sublimation;  but  more  direct  evidence 


CHAP.  XVII.]  Artificial  Minerals.  357 

is  afforded  by  the  presence  of  boracic  acid  in  the  condensed 
steam  issuing  from  volcanic  vents  and  furnaces  in  Tuscany 
and  other  places,  and  which  are  the  principal  source  of  sup- 
ply of  this  mineral  for  commercial  purposes.  By  the  solvent 
action  of  boracic  acid,  or  borax,  at  very  high  temperatures, 
many  refractory  or  ordinarily  infusible  substances  may  be 
made  to  combine  and  crystallise  from  fusion.  In  this  way, 
Ebelmen  produced  such  minerals  as  Spinel  (MgAiO4)  and 
Chrysoberyl  (BeAiO4)  from  mixtures  of  magnesia  and  alu- 
mina, and  glucina  and  alumina  respectively,  boracic  acid 
being  used  as  a  solvent.  The  colours  of  the  natural  minerals 
were  imitated  by  the  addition  of  oxide  of  chromium  for 
red,  iron  for  black,  and  cobalt  for  blue  spinel.  Alumina  was 
also  converted  into  crystals  of  corundum  and  ruby  by  heat- 
ing with  borax.  The  analogous  method  of  producing  crys- 
tallised titanic  acid  by  the  use  of  salt  of  phosphorus,  due  to 
Gustav  Rose,  has  already  been  noticed  at  page  312.  Hydro- 
fluoric acid  and  fluoride  of  silicon  have  also  been  used  to 
induce  combination  between  silica  and  metallic  oxides.  In 
this  way  Staurolite  has  been  formed  by  passing  hydrofluoric 
acid  through  alternating  layers  of  silica  and  alumina  and 
the  analogous  silicate  Topaz,  which  contains  fluoride  of  alu- 
minium, by  the  action  of  fluoride  of  silicon  upon  alumina. 

For  further  information  upon  this  most  interesting  class 
of  subjects  the  reader  is  referred  to  the  various  memoirs 
published  by  Ebelmen,  Daubree,  Senarmont,  and  others. 
A  compendious  notice  of  these  will  be  found  in  Percy's 
*  Swiney  Lectures  on  Geology,'  published  in  the  '  Chemical 
News,'  vol.  xxiv.,  and  Daubree,  '  Etudes  de  Geologic  Syn- 
thetique,'  Paris,  1879. 

Speaking  generally,  two  great  and  contrasted  groups  of 
causes  may  be  said  to  be  concerned  in  the  production  and 
modification  of  terrestrial  minerals.  These  are,  i.  The  pro- 
duction by  heat  within  the  crust  of  the  earth  of  homo- 
geneous silicates  of  the  alkaline  and  other  light  metals, 
analogous  to  glass.  Silicates  either  appear  at  the  surface 


358  Systematic  Mineralogy.         [CHAP.  XVII. 

by  eruption,  or  remain,  as  deep-seated  molten  masses,  to 
undergo  more  or  less  complete  devitrification  and  differentia- 
tion into  aggregates  of  quartz,  felspars,  and  other  silicates, 
phosphate  of  calcium,  magnetic  iron  ore,  metallic  sul 
phides ;  2.  The  conversion  of  these  aggregates  by  the 
action  of  air  and  carbonic  acid,  atmosphere  and  thermal 
waters,  into  the  various  classes  of  hydrated  silicates  and 
metallic  oxides,  alkaline  and  earthy  carbonates,  metallic 
sulphates,  &c.  These  actions  are  essentially  compensatory, 
the  tendency  of  the  second  being  towards  the  production  of 
quartz,  soluble  silica,  clay,  brown  iron  ore,  alkaline  carbonates, 
and  carbonate  of  lime ;  the  latter,  being  removed  in  solu- 
tion, can  only  be  returned  to  the  general  circulation  by  being 
brought  within  the  range  of  the  earth's  internal  heat.  That 
causes  of  this  kind  have  been  in  action  from  a  very  early 
period  of  the  earth's  existence  as  a  solid  body  is  evident 
from  the  identity  of  the  minerals  found  in  the  oldest  rocks 
with  those  recurring  in  other  places  in  similar  rocks,  which 
can  be  shown  to  have  been  formed  at  very  different  geo- 
logical periods. 

Daubree,  from  his  researches  on  the  synthesis  of  me- 
teorites, has  suggested  the  probability  of  an  earlier  period 
of  universal  scorification,  when  heavy  silicates,  such  as 
olivine  and  masses  of  the  heavier  metals,  may  have  been 
formed  within  the  crust  of  the  earth  and  depressed  below 
the  range  of  our  immediate  observation,  but  of  whose 
existence  evidence  is  furnished  by  their  presence  in  me- 
teorites and  volcanic  masses.  This  view  supposes  the 
formation  of  basic  silicates  by  the  action  of  heat  alone  to 
have  preceded  that  of  quartz,  and  the  silicates  of  the  felspar 
group,  where  the  intervention  of  water  is  assumed  to  be 
essential. 

^  Association  and  grouping  of  minerals.  This  subject  is 
intimately  connected  with  the  preceding,  as  it  is  only  by  a 
study  of  the  relative  positions  of  dissimilar  minerals  in  the 
same  aggregate  or  mass  that  their  order  of  succession  or 


CHAP.  XVLL]  Paragftusis.  359 

relative  ages  can  be  made  out.  The  hranrh  of  mineralogy 
devoted  to  this  class  of  inn^igatiqa^  and  which  *£xnfa  jn. 
very  close  relation  with  geology,  is  usually  known  as  Para 
genesis,  a  term  first  applied  by  Bretthaupt  in  1849,  in  a  work 
upon  the  associations  of  minerals  observed  in  the  veins 
worked  in  different  mining  Hi^iu-^,  The  farther  develop- 
ment of  the  same  class  of  observation  consequent  upon  the 
application  of  the  microscope  to  mmeialogkal  investigation, 
and  more  particularly  to  the  constitution  of  rock  ma^ape^  has 
given  rise  to  another  special  branch  known  as  Petrology, 
which  forms  the  subject  of  a  companion  volume  in  this! 


The  nature  of  these  associations  can  best  be  considered  in 
the  description  of  individual  minerals,  but  some  of  the  more 
prominent  groups  may  be  mentioned  here. 

Quartz  occurs  in  association  with  almost  every  other 
mineral,  but  is  more  commonly  found  together  with  ortfao- 
clase,  felspar,  and  other  so-called  acid  ghratt-^  rtian  with 
those  containing  less  silica.  It  is  also  very  frequently 
found  with  mica,  tourmaline,  ruble,  tin-stone,  topaz,  and  the 
richer  silver  ores.  Together  with  the  amorphous  varieties  of 
silica  (agate,  chalcedony,  &C.),  it  accompanies  hydrated 
silicates  of  the  zeotitic  group  in  basalt  and  vesicular  lavas, 
where  it  is  obviously  of  secondary  origin,  as  is  also  the  case 
when  it  occurs  in  mineral  veins  traversing  limestone  ^irata 

ZaAroribrzfr  (soda-lime  felspar)  is  almost  in  variably  found 
with  pyroxene,  hypersthene,  and  titamferous  iron  ore,  form- 
ing the  rocks  known  as  norite,  basalt.  &c. 

Of  die  different  minerals  containing  iron,  Magnetite  is 
commonly  associated  with  all  rocks  containing  fcuous  anH 
magnesium  silicates,  and  less  so  with  quartz  or  micaceous 
schists,  where  hematite  and  tuaniferous  iron  ores  are  mote 
generally  found.  Magnetite  occurring  in  large  maoc^ 
worked  for  iron  ores,  usually  contains  iron  pyrites,  chlorite, 
garnet,  hornblende,  and  apatite  in  small  quantities.  He- 
matite deposits  in  stratified  rocks  as  a  rule  r«ntain  as 
associates  quartz,  barytas,  fluorspar,  calcite.  and  aragonite. 


360  Systematic  Mineralogy.        [CHAP.  xvil. 

Spathic  iron  ore  is  generally  associated  with  sulphides,  such 
as  iron  pyrites,  copper  pyrites,  galena,  &c,  but  very  un- 
equally in  different  localities,  and  also  with  the  various  iso- 
morphous  carbonates  of  calcium,  zinc,  and  manganese, 
and  the  products  of  the  alteration  of  the  latter  such  as  pyro- 
lusite  (MnO2). 

Iron  Pyrites  is  the  most  abundant  of  all  the  metallic  sul- 
phides, and  is  widely  diffused  through  rocks  and  mineral 
deposits  of  all  kinds.  In  small  quantities  it  is  found  in  clays 
and  other  rocks  impermeable  to  water,  and  in  coal,  and  other 
carbonaceous  deposits,  where  it  is  protected  against  oxida- 
tion. When  in  large  masses,  it  is  commonly  associated 
with  copper  pyrites,  arsenical  pyrites,  and  the  various  sul- 
phides and  arsenides  of  nickel  and  cobalt,  gold  and  silver. 
It  also  forms  a  general  constituent  of  mineral  veins  con- 
taining the  ores  of  tin,  copper,  and  lead. 

Tinstone  and  its  associates  constitute  a  very  special 
group  of  minerals,  which,  though  restricted  to  a  small 
number  of  areas,  are  often  very  abundantly  developed  in 
particular  localities  within  these  areas.  In  this  group  are 
included  Tinstone  (Stannic  oxide),  tourmaline,  topaz, 
wolfram,  scheelite,  iron  pyrites,  mispickel,  copper  pyrites, 
chalcedony,  fluorspar,  hematite,  pitchblende,  and  occasion- 
ally bismuth  ores. 

Galena,  or  sulphide  of  lead,  the  principal  ore  of  that 
metal,  forms  part  of  numerous  well-marked  groups  of 
minerals,  which  as  a  rule  are  characteristic  of  particular 
districts.  Among  its  more  usual  associates  are  the  pro- 
ducts of  its  own  alteration,  sulphate,  carbonate,  and  phos- 
phate of  lead,  and  zincblende  (ZnS),  calamine  (carbonate 
and  silicate  of  zinc),  iron  and  copper  pyrites,  and  the 
*  waste '  or  earthy  minerals,  calcite,  fluorspar,  aragonite, 
dolomite,  and  barytes,  when  in  slate  and  limestone  districts  ; 
in  addition  to  which  quartz,  and  occasionally  zeolites,  are 
found  when  the  veins  are  in  siliceous  rocks,  such  as  granite, 
gneiss,  &c.  When  in  company  with  antimonial  minerals, 


CHAP.  XVII.]         Association  of  Minerals.  361 

gold  and  silver  ores  usually  enter  into  the  group,  as  in  the 
Hartz  and  Hungary. 

Nickel  and  cobalt  ores,  when  found  in  quantity,  i.e.  as 
rich  arsenides,  and  not  merely  as  mixtures  with  iron  pyrites, 
severally  accompany  native  arsenic  and  various  arsenides, 
the  products  of  their  oxidation  (pharmacolite,  cobalt  bloom, 
nickel  bloom),  and  the  different  ores  of  silver  and  bismuth. 

Copper  ores  are  very  commonly  found  in  connection 
with  minerals  containing  magnesia,  such  as  hornblende, 
chlorite,  serpentine,  and  dolomite,  and  also  with  quartz. 
Copper  pyrites,  the  most  abundant  ore  of  this  metal,  has 
two  principal  lines  of  association,  the  first  being  with  iron 
pyrites  in  more  or  less  intimate  mixture,  forming  the  so- 
called  coppery  pyrites,  and  the  second  with  copper-glance 
(Cu2S)  and  erubescite  (FeCu3S3),  forming  a  series  richer 
in  copper.  Native  copper,  and  the  various  oxides  and  oxy- 
salts,  carbonates,  sulphates,  and  phosphates  of  this  metal, 
are  common  products  of  the  alteration  of  the  sulphuretted 
minerals  ;  but  in  the  district  of  Lake  Superior  the  metal 
occurs  exceptionally  and  in  enormous  quantities  over  a  very 
large  area  practically  without  other  copper  ores,  and  in  as 
sociation  with  quartz,  calcite,  and  zeolitic  minerals. 

Rock  Salt  (NaCl),  though  often  found  in  very  large 
masses  in  a  perfectly  pure  state,  is  generally  associated  with 
salts  of  calcium  and  the  alkaline  metals,  especially  gypsum 
and  anhydrite.  Where  the  deposits  have  been  isolated  in 
such  a  manner  that  the  more  soluble  salts  contained  in  the 
original  salt  water  have  been  preserved,  a  numerous  series  of 
double  sulphates  and  chlorides  of  potassium,  magnesium, 
&c.,  are  developed.  Among  these  are  Kainite.  Carnallite, 
Kaluszite,  Boracite,  Polyhalite,  &c.,  besides  bromide  of 
magnesium.  This  is  only  seen  on  a  very  large  scale  at 
Stassfurt  in  Prussia,  and  Kalusz  in  Gallicia. 

As  regards  the  frequency  or  scarcity  of  the  occurrence 
of  minerals,  it  may  be  useful  to  remember  that  a  substance 
may  be  rare  in  two  different  ways — being  either  widely  dis- 


362  Systematic  Mineralogy.        [CHAP.  XVIL 

tributed,  but  in  minute  or  even  invisible  quantities  in  other 
minerals,  or  restricted  to  a  few  localities,  where,  however,  it 
may  be  found  isolated  in  quantity.  Sulphide  of  cadmium 
may  be  taken  as  an  instance  of  the  first  kind,  having  only 
been  found  in  a  single  locality  as  an  independent  mineral 
(Greenockite),  and  in  a  few  minute  examples;  but  as  an 
isomorphous  associate  with  the  corresponding  sulphide  of 
zinc,  it  is  present  in  the  larger  number  of  samples  of  zinc- 
blende,  so  that  some  tons  of  the  metal  are  made  annually 
by  fractional  distillation  of  the  first  deposits  of  zinc  oxide 
obtained  in  the  zinc  works,  which  are  usually  found  to  be 
cadmiferous.  Other  examples  are  afforded  by  the  rare  alka- 
line metals  coesium  and  rubidium,  common  in  certain 
mineral  waters,  but  almost  unknown  in  individual  minerals  ; 
also  by  thallium,  gallium,  and  similar  elements  existing  in 
spectroscopic  traces  in  common  minerals  such  as  pyrites, 
zincblende,  &c.  Cryolite  (6NaFl  +  A1F16)  is  an  example  of 
the  second  kind  of  rarity,  it  being  almost  entirely  restricted  to 
one  spot  on  the  coast  of  Greenland,  where,  however,  it  is 
found  in  such  masses  that  it  can  be  utilised  as  a  commercial 
source  of  soda  and  alum. 


INDEX- 


ACT 

A  CICULAR  crystals,  188 
XTL     Acids,  327 
Airy's  spirals,  256 
Albite,  forms  of,  162 

—  twin  forms  of,  183 
Allanite,  form  of,  155 
Allochromatic  minerals,  283 
Alteration  of  minerals,  345 
Alum  group,  337 
Alumina,  tests  for,  309,  316 
Analogue  poles,  297 
Angle  of  optic  axes,  263 
Anglesite,  form  of,  140-141 
Anharmonic  property  of  zones,  34 

—  ratios  of  planes,  32 
Anisometric  projection,  195 
Anisotropic  media,  225 
Anorthic  system,  156 
Antilogue  poles,  297 
Antimonic  oxide,  -dimorphous,  334 
Apatite,  pyramidal  hemihedron  of,  104 
Apatite  group  of  minerals,  336 
Aragonite,  twin  forms  of,  177 
Association  of  minerals,  358 
Asterism,  289 

Asymmetric  system,  156 

Atacamite,  production  of,  349 

Athermancy,  292 

Atomic  weight,  323 

Atom?,  323 

Augite  group,  339 

Augite,  as  a  furnace-product,  354 

Avanturine,  289 

Avpgadro's  law,  323 

Axinite,  forms  of,  162 

Axis,  optic,  238 

—  angle  of,  263 
Azurite,  form  of,  155 


TDABINET'S  goniometer,  193 
J3     Babingtonite,  forms  of,  162 
Baryta,  test  for,  317 
Barytes,  form  of,  139,  140 
Bases,  329 


CLE 

Baveno  type  of  twin  crystal,  182 
Biaxial  crystals,  257 

—  principal  sections  of,  244 

—  wave-surface,  241 
Binary  system,  145 
Bisectrices  of  optic  axes,  242 
Blowpipe,  299 

Borax,  form  of,  155 

—  as  a  blowpipe  flux,  309 
Botryoidal  aggregates,  189 
Brachydiagonal  axis,  129,  156 

—  quarter  pyramids,  158 
Brachy  dome,  134 

—  pinakoid,  134,  160 
Brayais-Miller  hexagonal  notation,  75 
Breithaupt  on  Paragenesis,  359 
Breon's  method  of  separating  minerals, 

217 

Brezina's  plate,  259 
Brittleness,  212 
Brookite,  138 
Brushite,  form  of,  154 


/^ALCIFERRITE,  338 
\~s     Calcite,   combinations  of,   96,  97 
99,  100 

—  irregular  aggregates  of,  187 

—  twin  crystals  of,  172,  173,  174 
Caledonite,  forms  of,  154 
Carbon,  334 

Carbonate  of  lime,  dimorphous,  334 
Carlsbad  type  of  t\vin  crystal,  181 
Chemical  constitution,  320 

—  properties  of  minerals,  298 
Chrysolite,  as  a  furnace-product,  334 
Circular  polarisation,  254 
Cleavage,  205 

—  qualities  of,  207 
Cleavages,  principal  : — 

Cubic  system,  207 
Hexagonal  system,  208 
Tetragonal  system,  208 
Rhombic  system,  208 
Oblique  system,  208 
Triclinic  system,  209 


364 


Index. 


CLI 

Clinoclase,  forms  of,  154 
Clinodiagonal  axis,  147 
Cljnodpmes,  150 
Clinopinakoid,  151 
Clinorhombic  system,  145 
Colour,  280 
Columnar  crystals,  188 
Combinations  of  cubic  syst'  m,  63 
Constants,  optical,  277 
Contact  goniometer,  191 

—  twins,  1 68 

Conversion  of  notation,  so 
Copper,  tests  for,  319 

—  glance,  twin  forms  of,  178 

—  ores,  occurrence  of,  361 

—  pyrites,  forms  of,  125 
Corundum  group,  336 
Crookes  on  Phosphoresence,  291 
Crossed  dispersion,  269 
Cryolite,  rarity  of,  362 
Crystal,  etymology  of,  7 
Cube,  37,  48 

—  combinations  of,  64,  65,  67,  68,  69,  70, 
72,  73 

—  hemihedral  forms  of,  58 

—  twin  crystals,  169 

Cubic  system,  symmetry  of,  37 

—  general  form  of,  39 
Curved  faces  of  crystals,  188 

DAUBREE  on  origin  of  minerals, 
353.  356 

Deltoid-dodecahedron,  57 
Density,  213 

Deviation,  minimum,  232 
Diamond,  curved  crystals  of,  188 
Diaspore  group,  337 
Diathermancy,  292 
Dichroiscope,  285 
Dichroism,  284 
Dihexagonal  prism,  82 

—  pyramid,  combinations  of,  78 
Dimorphism,  333 

Disperson  of  optic  axes,  265 

—  of  the  median  lines,  267 
Ditetragonal  prism,  116 

—  pyramid,  113 
Ditrigonal  prism,  107 
Double  refraction,  237 

—  determination  of  sign,  270 
Doubly-oblique  system,  156 
Ductility,  212 
Dyakisdodecahedron,  53 

—  combinations  of,  71,  72 

"T*  BELMEN'S     artificial     minerals, 

S*f  ,  357 

Einghedng,  156 

Elasticity,  212 

Electric  calamine,  forms  of,  165 

—  twin  forms  of,  179 

Electrical  properties  of  minerals.  296 


HEX 

Equivalents,  chemical,  321 
Essential  qualities  of  minerals,  2 
Exner  on  hardness  in  different  directions, 
211 


FACE  determined  by  two  zones,  31 
Faces,  geometrical  relations  of,  24 
Felspar  as  a  furnace-product,  354 
Fibrous  aggregates,  189 
First  median  line,  242 
Flame  reactions,  304 
Flexibility,  212 
Flos  ferri,  189 

Fluorspar,  irregular  forms  of,  187 
—  twin  crystals  of,  169 
Fluorescence,  200 
Formulae,  chemical,  326 
Fracture,  209 
Fuess's  goniometer,  193 
Furnace  products,  354 
Fusibility,  303 


/"*  ALENA,  as  a  furnace-product,  355 

v_T    —  association  of,  360 

Garnet  group,  340 

Gold,  tests  for,  319 

Goniometer,  191 

Graphite,  artificial,  356 

Greenockite,  forms  of,  165 

—  rarity  of,  362 
Gypsum,  alteration  of,  346 

—  curved  crystals  of,  188 

—  twin  forms  of,  180 


HABIT  of  crystals,  188 
Haidinger's  dichroiscope,  285 
Hardness,  210 

Hausmannite,  twin  crystals  of,  175 
Hematite,  association  of,  360 
Hemi-brachydome,  160 
Hemihedral  cubic  combinations,  70 

—  forms,  13 

—  cubic  forms,  49 

—  cubic  diagrams,  S9 
Hemimacrodomes,  159 
Hemimorphism,  163 
Hemiorthodomes,  130 
Hemitrope  crystals,  168 
Heteromorphism,  333 
Heterotropic  media,  225 
Hexagonal  axes,  projection  of,  196 
— •  basal  pinakoid,  84 

—  hemihedral  forms,  88 

—  holohedral  combinations,  85 
diagrams,  84 

—  Weiss's  notation,  74 

—  prism,  83 

—  of  second  order,  83 

—  prism,  twin  crystals  of,  173,  174 

—  pyramid,  80 


Index. 


365 


HEX 

Hexagonal  pyramid  of  second  order,  8r 

—  pyramid  of  third  order,  103 

—  scalenohedra,  92 

—  symmetry,  73 

—  tetartohedra,  104 

—  trapezohedral  hemihedra,  91 

—  twin  crystals,  172 
Hexakisoctahedron,  39 
Hexakisoctahedron,  combinations  of,  69, 

70 

Hexakistetrahedron,  55 
Hirschwald's  goniometer,  184 
Holohedral  cubic  diagram,  49 

—  forms,  13 

—  tetragonal  combinations,  118 

—  tetragonal  diagram,  117 
Homeomorphism,  336 
Horizontal  dispersion,  269 
Hornblende,  forms  of,  155 
Hydrochloric  acid,  test  for,  319 

TCOSITETRAHEDRON,  41 
-L     —  combinations  of,  65,  66,  67 
Ideochromatism,  283 
Ilmenite,  crystalline  form  of,  no 
Imperfections  of  crystals,  186 
Inclined  dispersion,  268 
Inclined  hemihedra,  cubic,  55 
Inclosures  in  minerals,  275 
Index  of  refraction,  230 

—  determination  of,  234 
Interference  figures,  260 

—  waves,  222 
Iridescence,  288 
Iron,  reactions  of,  316 
Irregular  groups  of  crystals,  189 
— •  polarisation,  273 
Isochromatic  curves,  253 
Isometric  projection,  195 
Isomorphism,  335 

Isotropic  media,  225 


TOLY'S  spring  balance,  216 


T    ABRADORITE,  association  of,  359 
1  -*    —  twin  striation  in,  183 
Laumonite,  alteration  of,  345 
Lead,  tests  for,  318 
Levy's  notation,  23 
Lime,  tests  for,  316 
Limonite,  formation  of,  352 
Linear  projection,  200 
Lustre,  286 


MACLED  crystals,  168 
Macrodiagonal  axis,  129,  156 
—  quarter  pyramid,  158 
Macrodome,  133 
Macro  pinakoid,  134,  160 


PAR 

Magnesia,  tests  for,  309,  316 
Magnetism,  297 
Magnetite,  association  of,  359 
Mallard  on  polysynthetic  structure,  274 
Malleability,  212 
Mammillary  aggregates,  189 
Manebach  type  of  twin  crystal,  182 
Median  lines  of  optic  axes,  242 
Mica,  asterism  of,  290 
Miller's  notation,  18 

—  rhombohedral  notation,  74 
Minimum  deviation,  233 
Mitscherlich's  goniometer,  193 
Mitscherlich  on  isomorphism,  335 
Mohr's  method  of  gauging,  215 
Mohs'  scale  of  hardness,  210 
Molecules,  323 

—  physical,  203 
Molybdates,  forms  of,  127 
Monoclinic  system,  145 
Monodimetric  projection,  195 
Monosymmetric  hemihedrism,  144 
Monosymmetry,  152 


AT  AUM ANN'S  notation,  21 

IN      Negative  biaxial  crystals,  243 

Nicol's  prism,  247 

Nitrate  of  barium,  distorted  forms  of,  185 

Nodules,  189 

Notation  of  crystals,  16 


OBLIQUE  axes,  projection  of,  197 
—  hemipyramids,  147 

—  prism,  149 

—  rhombic  pyramid,  147 

—  rhomboidal  pyramid,  157 

—  system,  145 

—  twin  crystals,  180 
Octahedron,  44 

—  combinations  of,  62,  63,  64,  63,  66,  6 
69;  71 

• —  distorted  forms  of,  184 
• —  twin  crystals,  169 
Olivine,  forms  of,  142 
Optic  axis,  238 

—  angle  of,  263 
Optical  classification,  245 

—  constants,  277 

—  properties  of  minerals,  218 
Origin  of  minerals,  353 
Orthoclase,  twin  forms  of,  181 
Orthodiagonal  axis,  147 
Orthohexagonal  notation,  74 
Orthopinakoid,  151 


pARAGENESIS,  359 
JT      Parallel  grouping  of  crystals,  165 
Parallel  hemihedra,  cubic,  52 
Parameters  of  hexagonal  pyramid,  deter- 
mination of,  79 
—  rhombohedron,  determination  of,  101 


366 


Indi 


'ex. 


PAR 

Parameters  of  tetragonal  pyramid,  deter- 
mination of,  114 
Penetration  twins,  168 
Pentagonal  dodecahedron,  54 

—  combinations,  70 

—  tetartohedral,  61 

—  twin  crystals  of,  170 
Phosphorescence,  290 
Phosphoric  acid,  test  for,  320 
Physiography,  3 
Plagihedra   cubic,  51 
Plane  of  composition,  167 

—  of  contact,  167 
Pleochroism,  283 
Polariscope,  249 
Polysymmetry,  339 
Polysynthetic  crystals,  274 
Positive  biaxial  crystals,  243 
Projections,  elements  of,  196 

—  perspective,  195 
Pseudomorphism,  349 

Pyramidal  hemihednsm,  hexagonal,  103 
Pyrites,  alteration  of,  352 

—  twin  crystals  of,  170 
Pyroelectricity,  297 

VUANTIVALENCE,  326 
/     Quartz,  association  of,  359 
distorted  crystals  of,  185 

—  double  rotation  of,  256 

—  right  and  left-handed  crystals  of,  255 

—  tetartohedron  of,  108 

—  twin  crystals  of,  175 
Quenstedt  s  projection,  200 

T)  ATIONALITY,  principle  of,  8 

£v     Reflected  waves,  227 
Reflecting  goniometer,  191 
Refraction,  228 

—  double,  237 

Regular  solids  possible  as  crystals,  9 
Reniform  aggregates,  189 
Reticular  point  systems,  14 
Rhombic  combinations,  136 

—  dodecahedron,  47 

combinations  of,  64,  66,  67,  68,  70 

72 

distorted  forms  of,  185 

hemihedral  forms  of,  58 

-  twins  of,  169 

—  forms,  diagram  of,  134 
— •  hemihedral  forms,  143 

—  prism,  131 

—  pyramid,  129 

determination  of  parameters  of,  131 

—  sphenoids,  143 

—  symmetry,  128 

—  twin  crystals,  177 
Rhombohedra,  normal  or  hemihedral,  04 

—  of  second  ordw,  no 

—  of  third  order,  109 

—  of  longer  polar  edges  of,  scalenohedra, 


Rhombohedra  of  middle  edges  of  scaleno- 
hedra, 97 

—  of  shorter  polar  edges  of  scalenohe- 
dra, 98 

—  limits  of,  96 

—  tetartohedral,  109 

—  twin  crystals  of,  172,  173,  174 
Rhombohedral  combinations,  95,  96,  97, 

99, 100 

—  of  positive  and  negative,  95 

—  notation,  Miller's,  in 

—  Naumann's,  101 

Right  and  left  handed  rotation,  255 
Rock-salt,  association  of,  361 
Rocks,  definition  of,  5 
Rose,  G.,  on  titanic  acid,  312 

—  on  tridymite,  312 

—  on  carbonate  of  lime,  335 
Rotatory  power  of  quartz,  254 

—  of  cinnabar,  255 
Rutile,  twin  crystals  of,  176 

SALTS,  330 
Satin  spar,  189 
Scale  of  hardness,  210 
Scalenohedra,  limits  of,  97 

—  twin  crystals  of,  173,  174 
Schraufs'  hexagonal  notation,  74 
Senarmont's  experiments,  292 
Siderite,  curved  crystals  of,  188 
Silica,  allotropic,  334 

Silver,  tests  for,  318 
Sodalite,  twin  crystals  of,  169 
Sonstadt's  solution,  217 
Sorby's  method  of  finding  index   of  re- 
fraction, 234 
Species,  definition  of,  3 
Specific  gravity,  213 
Sphenoidal  combinations,  125 

—  twin  crystals,  176 
Spherical  projection,  201 
Spinel,  twin  crystals  of,  171 
--  group,  337 
Stalactites,  189 
Stalagmites,  189 

Star  sapphire,  289 
Staurohte  twin  forms,  178 
Stauroscope,  258 
Streak,  283 

Striations  on  quartz  crystals    187 
Struvite,  forms  of,  165 
Sulphate  of  magnesium,  144 
Sulphur,  allotropic,  333 

—  forms  of.  136,  137 
Sulphuric  acid,  test  for,  319 
Symmetry,  axes  of,  10 

—  of  crystals,  8 

—  crystals  classified  by,  n 

—  limitation  of  possible, 

"PABULAR  crystals,  iss 

-L      Tautozonahty,  29 

—  condition  of,  30 


Index. 


367 


TEN 

Tenacity,  212 
Tetartohedral  forms,  13 
Tetartohedron,  cubic,  60 
Tetarto-pyramids,  157 
Tetragonal,  basal  pinakoid.  117 

—  hemihedrism,  120 

—  prisms  of  first  order,  116 

—  pyramid,  114 

—  pyramidal  hemihedrism,  126 

—  pyramids  of  second  order,  115 

—  pyramids  of  third  order,  126 

—  pyramids,  limits  of,  116 

—  pyramids,  twin  crystals  of,  175 

—  scalenohedra,  123 

—  sphenoids,  >  24 

—  symmetry,  1 12 

—  trapezohedra,  121,  127 

—  twin  crystals,  175 
Tetrahedron,  57 

—  combinations,  72,  73 

—  twin  crystals  of,  169-170 
Tetrakishexahedron,  45 

—  combinations  of,  67,  68,  69 
Thermal  relations  of  minerals,  291 
Thermoelectricity,  297 
Tinstone,  association  of,  360 

—  twin  ciystals  of,  176 
Titanic  acid,  trimorphous,  334 
Topaz,  137 ' 

Total  reflection,  231 
Toughness,  212 
Tourmaline,  forms  of,  164 
Translucency,  280 

Trapezohedral,  tetartohedrism  hexago- 
nal, 105 

Triakisoctahedron,  43 
— •  combinations  of,  66,  67 
Triakistetrahedron,  £.6 


7.WE 

Triclinic  axes,  projection  of,  15 

—  hemiprisms,  159 

—  quarter  pyramids,  157 
— •  system,  156 

—  twin  crystals,  183 
Trigonal  pyramids,  106 
Tungstates,  forms  of,  127 
Twin  axis,  167 

—  grouping,  166 

—  plane,  167 


T  TNI  AXIAL  crystals,  238 
\J     —  behaviour  in  parallel  polarised 

light,  250 
in  convergent  polarised  light,  251 


VON  KOBELL'S  scale  of  fusibility, 
3°3 
—  stauroscope,  258 


WAVE  motion,  219 
Weiss's  notation,  17 
White  lead  ore,  twin  forms  of,  177 
Witherite,  forms  of,  143 
Wollaston's  goniometer,  191 


^EOLITES,  formation  of,  353 
fi-j     Zinc,  test  for,  309 
Zone  axis,  29 

—  plane,  30 

—  symbol,  determination  of,  30 
Zones,  29 

Zwei-  und  eingliedrig,  145 


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